System and method for tranmissions using eliptical core fibers

ABSTRACT

A system for transmission of optical data signals has first optical processing circuitry for receiving a plurality of digital signals and applying at least one of a Hermite-Gaussian function, a Laguerre-Gaussian function or an Ince-Gaussian function to each of the received plurality of digital signals. The first optical processing circuitry also combines each of the at least one of the Hermite-Gaussian function, the Laguerre-Gaussian function or the Ince-Gaussian function applied plurality of digital signals into a single carrier signal. An optical transmitter transmits the single carrier signal. An optical receiver receives the transmitted single carrier signal. Second optical processing circuitry separates the at least one of the Hermite-Gaussian function, the Laguerre-Gaussian function or the Ince-Gaussian function applied digital signals of the single carries signal into separate signals and removes the at least one of the Hermite-Gaussian function, the Laguerre-Gaussian function or the Ince-Gaussian function applied to each of the plurality of digital signals. An elliptical core fiber transmits the single carrier signal from the optical transmitter to the optical receiver. The elliptical core fiber includes an elliptical core have a major axis and a minor axis.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No.62/342,693, filed on May 27, 2016, entitled SYSTEM AND METHOD FORTRANMISSIONS USING ELIPTICAL CORE FIBERS (Atty. Dkt. No. NXGN-33141).This application also claims the benefit of U.S. Provisional ApplicationNo. 62/343,505, filed on May 31, 2016, entitled SYSTEM AND METHOD FORTRANMISSIONS USING ELIPTICAL CORE FIBERS (Atty. Dkt. No. NXGN-33151).This application also claims the benefit of U.S. Provisional ApplicationNo. 62/362,915, filed on Jul. 15, 2016, entitled SYSTEM AND METHOD FORTRANMISSIONS USING ELIPTICAL CORE FIBERS (Atty. Dkt. No. NXGN-33209).U.S. Provisional Application Nos. 62/342,693, 62/343,505 and 62/362,915are incorporated by reference in their entirety. This application isalso a Continuation-in-Part of U.S. patent application Ser. No.14/882,085, filed on Oct. 13, 2015, entitled APPLICATION OF ORBITALANGULAR MOMENTUM TO FIBER, FSO AND RF (Atty. Dkt. No. NXGN-32777) whichis incorporated herein by reference in its entirety.

TECHNICAL FIELD

The present invention relates to optical signal transmissions, and moreparticularly, to the use of elliptical core fibers for the transmissionof optical signals.

BACKGROUND

Optical communications may be carried out over an optical fiber usingoptical signals processed with orthogonal functions such asHermite-Gaussian functions and Laguerre-Gaussian functions as describedhereinabove in order to improve system bandwidth. Laguerre-Gaussian (LG)and Hermite-Gaussian (HG) signals have three important properties andadvantages and one major disadvantage. The advantages include theability to form two complete families of exact and orthogonal solutionsof the paraxial wave equations. Another advantage is that the HG and LGsignals are transverse eigenmodes of stable resonators. Finally, the HGand LG signals do not change shape on propagation and provide stablemodes of propagation for signals. The disadvantage of HG and LG signalsis that the eigenmodes coupled to one another after a long distance ofpropagation. However, for short distance applications such as intra-dataand inter-data center connectivity, and front haul applications or backhaul applications, both LG and HG signals provide good solutions withinthese applications. Thus, there is a need for a manner of transmittingoptical signals over a fiber that limits the coupling issues for usewith longer distance fiber tranmissions.

SUMMARY

The present invention, as disclosed and describe herein, in one aspectthereof, comprises a system for transmission of optical data signals hasfirst optical processing circuitry for receiving a plurality of digitalsignals and applying at least one of a Hermite-Gaussian function, aLaguerre-Gaussian function or a Ince-Gaussian function to each of thereceived plurality of digital signals. The first optical processingcircuitry also combines each of the at least one of the Hermite-Gaussianfunction, the Laguerre-Gaussian function or the Ince-Gaussian functionapplied plurality of digital signals into a single carrier signal. Anoptical transmitter transmits the single carrier signal. An opticalreceiver receives the transmitted single carrier signal. Second opticalprocessing circuitry separates the at least one of the Hermite-Gaussianfunction, the Laguerre-Gaussian function or the Ince-Gaussian functionapplied digital signals of the single carries signal into separatesignals and removes the at least one of the Hermite-Gaussian function,the Laguerre-Gaussian function or the Ince-Gaussian function applied toeach of the plurality of digital signals. An elliptical core fibertransmits the single carrier signal from the optical transmitter to theoptical receiver. The elliptical core fiber includes an elliptical corehave a major axis and a minor axis.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding, reference is now made to thefollowing description taken in conjunction with the accompanyingDrawings in which:

FIG. 1 illustrates various techniques for increasing spectral efficiencywithin a transmitted signal;

FIG. 2 illustrates a particular technique for increasing spectralefficiency within a transmitted signal;

FIG. 3 illustrates a general overview of the manner for providingcommunication bandwidth between various communication protocolinterfaces;

FIG. 4 illustrates the manner for utilizing multiple level overlaymodulation with twisted pair/cable interfaces;

FIG. 5 illustrates a general block diagram for processing a plurality ofdata streams within an optical communication system;

FIG. 6 is a functional block diagram of a system for generating orbitalangular momentum within a communication system;

FIG. 7 is a functional block diagram of the orbital angular momentumsignal processing block of FIG. 6;

FIG. 8 is a functional block diagram illustrating the manner forremoving orbital angular momentum from a received signal including aplurality of data streams;

FIG. 9 illustrates a single wavelength having two quanti-spinpolarizations providing an infinite number of signals having variousorbital angular momentums associated therewith;

FIG. 10A illustrates an object with only a spin angular momentum;

FIG. 10B illustrates an object with an orbital angular momentum;

FIG. 10C illustrates a circularly polarized beam carrying spin angularmomentum;

FIG. 10D illustrates the phase structure of a light beam carrying anorbital angular momentum;

FIG. 11A illustrates a plane wave having only variations in the spinangular momentum;

FIG. 11B illustrates a signal having both spin and orbital angularmomentum applied thereto;

FIGS. 12A-12C illustrate various signals having different orbitalangular momentum applied thereto;

FIG. 12D illustrates a propagation of Poynting vectors for various Eigenmodes;

FIG. 12E illustrates a spiral phase plate;

FIG. 13 illustrates a system for using to the orthogonality of an HGmodal group for free space spatial multiplexing;

FIG. 14 illustrates a multiple level overlay modulation system;

FIG. 15 illustrates a multiple level overlay demodulator;

FIG. 16 illustrates a multiple level overlay transmitter system;

FIG. 17 illustrates a multiple level overlay receiver system;

FIGS. 18A-18K illustrate representative multiple level overlay signalsand their respective spectral power densities;

FIG. 19 illustrates comparisons of multiple level overlay signals withinthe time and frequency domain;

FIG. 20A illustrates a spectral alignment of multiple level overlaysignals for differing bandwidths of signals;

FIG. 20B-20C illustrate frequency domain envelopes located in separatelayers within a same physical bandwidth;

FIG. 21 illustrates an alternative spectral alignment of multiple leveloverlay signals;

FIG. 22 illustrates three different superQAM signals;

FIG. 23 illustrates the creation of inter-symbol interference inoverlapped multilayer signals;

FIG. 24 illustrates overlapped multilayer signals;

FIG. 25 illustrates a fixed channel matrix;

FIG. 26 illustrates truncated orthogonal functions;

FIG. 27 illustrates a typical OAM multiplexing scheme;

FIG. 28 illustrates various manners for converting a Gaussian beam intoan OAM beam;

FIG. 29A illustrates a fabricated metasurface phase plate;

FIG. 29B illustrates a magnified structure of the metasurface phaseplate;

FIG. 29C illustrates an OAM beam generated using the phase plate withl=+1;

FIG. 30 illustrates the manner in which a q-plate can convert a leftcircularly polarized beam into a right circular polarization orvice-versa;

FIG. 31 illustrates the use of a laser resonator cavity for producing anOAM beam;

FIG. 32 illustrates spatial multiplexing using cascaded beam splitters;

FIG. 33 illustrated de-multiplexing using cascaded beam splitters andconjugated spiral phase holograms;

FIG. 34 illustrates a log polar geometrical transformation based on OAMmultiplexing and de-multiplexing;

FIG. 35 illustrates an intensity profile of generated OAM beams andtheir multiplexing;

FIG. 36A illustrates the optical spectrum of each channel after eachmultiplexing for the OAM beams of FIG. 10A;

FIG. 36B illustrates the recovered constellations of 16-QAM signalscarried on each OAM beam;

FIG. 37A illustrates the steps to produce 24 multiplex OAM beams;

FIG. 37B illustrates the optical spectrum of a WDM signal carrier on anOAM beam;

FIG. 38A illustrates a turbulence emulator;

FIG. 38B illustrates the measured power distribution of an OAM beamafter passing through turbulence with a different strength;

FIG. 39A illustrates how turbulence effects mitigation using adaptiveoptics;

FIG. 39B illustrates experimental results of distortion mitigation usingadaptive optics;

FIG. 40 illustrates a free-space optical data link using OAM;

FIG. 41A illustrates simulated spot sized of different orders of OAMbeams as a function of transmission distance for a 3 cm transmittedbeam;

FIG. 41B illustrates simulated power loss as a function of aperturesize;

FIG. 42A illustrates a perfectly aligned system between a transmitterand receiver;

FIG. 42B illustrates a system with lateral displacement of alignmentbetween a transmitter and receiver;

FIG. 42C illustrates a system with receiver angular error for alignmentbetween a transmitter and receiver;

FIG. 43A illustrates simulated power distribution among different OAMmodes with a function of lateral displacement;

FIG. 43B illustrates simulated power distribution among different OAMmodes as a function of receiver angular error;

FIG. 44 illustrates a bandwidth efficiency comparison for square rootraised cosine versus multiple layer overlay for a symbol rate of ⅙;

FIG. 45 illustrates a bandwidth efficiency comparison between squareroot raised cosine and multiple layer overlay for a symbol rate of ¼;

FIG. 46 illustrates a performance comparison between square root raisedcosine and multiple level overlay using ACLR;

FIG. 47 illustrates a performance comparison between square root raisedcosine and multiple lever overlay using out of band power;

FIG. 48 illustrates a performance comparison between square root raisedcosine and multiple lever overlay using band edge PSD;

FIG. 49 is a block diagram of a transmitter subsystem for use withmultiple level overlay;

FIG. 50 is a block diagram of a receiver subsystem using multiple leveloverlay;

FIG. 51 illustrates an equivalent discreet time orthogonal channel ofmodified multiple level overlay;

FIG. 52 illustrates the PSDs of multiple layer overlay, modifiedmultiple layer overlay and square root raised cosine;

FIG. 53 illustrates a bandwidth comparison based on −40 dBc out of bandpower bandwidth between multiple layer overlay and square root raisedcosine;

FIG. 54 illustrates equivalent discrete time parallel orthogonalchannels of modified multiple layer overlay;

FIG. 55 illustrates four MLO symbols that are included in a singleblock;

FIG. 56 illustrates the channel power gain of the parallel orthogonalchannels of modified multiple layer overlay with three layers andT_(sym)=3;

FIG. 57 illustrates a spectral efficiency comparison based on ACLR1between modified multiple layer overlay and square root raised cosine;

FIG. 58 illustrates a spectral efficiency comparison between modifiedmultiple layer overlay and square root raised cosine based on OBP;

FIG. 59 illustrates a spectral efficiency comparison based on ACLR1between modified multiple layer overlay and square root raised cosine;

FIG. 60 illustrates a spectral efficiency comparison based on OBPbetween modified multiple layer overlay and square root raised cosine;

FIG. 61 illustrates a block diagram of a baseband transmitter for a lowpass equivalent modified multiple layer overlay system;

FIG. 62 illustrates a block diagram of a baseband receiver for a lowpass equivalent modified multiple layer overlay system;

FIG. 63 illustrates a channel simulator;

FIG. 64 illustrates the generation of bit streams for a QAM modulator;

FIG. 65 illustrates a block diagram of a receiver;

FIG. 66 is a flow diagram illustrating an adaptive QLO process;

FIG. 67 is a flow diagram illustrating an adaptive MDM process;

FIG. 68 is a flow diagram illustrating an adaptive QLO and MDM process

FIG. 69 is a flow diagram illustrating an adaptive QLO and QAM process;

FIG. 70 is a flow diagram illustrating an adaptive QLO, MDM and QAMprocess;

FIG. 71 illustrates the use of a pilot signal to improve channelimpairments;

FIG. 72 is a flowchart illustrating the use of a pilot signal to improvechannel impairment;

FIG. 73 illustrates a channel response and the effects of amplifiernonlinearities;

FIG. 74 illustrates the use of QLO in forward and backward channelestimation processes;

FIG. 75 illustrates the manner in which Hermite Gaussian beams andLaguerre Gaussian beams diverge when transmitted from phased arrayantennas;

FIG. 76A illustrates beam divergence between a transmitting aperture anda receiving aperture;

FIG. 76B illustrates the use of a pair of lenses for reducing beamdivergence;

FIG. 77 illustrates the configuration of an optical fiber communicationsystem;

FIG. 78A illustrates a single mode fiber;

FIG. 78B illustrates multi-core fibers;

FIG. 78C illustrates multi-mode fibers;

FIG. 78D illustrates a hollow core fiber;

FIG. 79 illustrates the first six modes within a step index fiber;

FIG. 80 illustrates the classes of random perturbations within a fiber;

FIG. 81 illustrates the intensity patterns of first order groups withina vortex fiber;

FIGS. 82A and 82B illustrate index separation in first order modes of amulti-mode fiber;

FIG. 83 illustrates a few mode fiber providing a linearly polarized OAMbeam;

FIG. 84 illustrates the transmission of four OAM beams over a fiber;

FIG. 85A illustrates the recovered constellations of 20 Gbit/sec QPSKsignals carried on each OAM beam of the device of FIG. 84;

FIG. 85B illustrates the measured BER curves of the device of FIG. 84;

FIG. 86 illustrates a vortex fiber;

FIG. 87 illustrates intensity profiles and interferograms of OAM beams;

FIG. 88 illustrates a free-space communication system;

FIG. 89 illustrates a block diagram of a free-space optics system usingorbital angular momentum and multi-level overlay modulation;

FIGS. 90A-90C illustrate the manner for multiplexing multiple datachannels into optical links to achieve higher data capacity;

FIG. 90D illustrates groups of concentric rings for a wavelength havingmultiple OAM valves;

FIG. 91 illustrates a WDM channel containing many orthogonal OAM beams;

FIG. 92 illustrates a node of a free-space optical system;

FIG. 93 illustrates a network of nodes within a free-space opticalsystem;

FIG. 94 illustrates a system for multiplexing between a free spacesignal and an RF signal;

FIG. 95 illustrates a seven dimensional QKD link based on OAM encoding;

FIG. 96 illustrates the OAM and ANG modes providing complementary 7dimensional bases for information encoding;

FIG. 97 illustrates a block diagram of an OAM processing systemutilizing quantum key distribution;

FIG. 98 illustrates a basic quantum key distribution system;

FIG. 99 illustrates the manner in which two separate states are combinedinto a single conjugate pair within quantum key distribution;

FIG. 100 illustrates one manner in which 0 and 1 bits may be transmittedusing different basis within a quantum key distribution system;

FIG. 101 is a flow diagram illustrating the process for a transmittertransmitting a quantum key;

FIG. 102 illustrates the manner in which the receiver may receive anddetermine a shared quantum key;

FIG. 103 more particularly illustrates the manner in which a transmitterand receiver may determine a shared quantum key;

FIG. 104 is a flow diagram illustrating the process for determiningwhether to keep or abort a determined key;

FIG. 105 illustrates a functional block diagram of a transmitter andreceiver utilizing a free-space quantum key distribution system;

FIG. 106 illustrates a network cloud-based quantum key distributionsystem;

FIG. 107 illustrates a high-speed single photon detector incommunication with a plurality of users; and

FIG. 108 illustrates a nodal quantum key distribution network.

FIG. 109 illustrates the use of a reflective phase hologram for dataexchange;

FIG. 110 is a flow diagram illustrating the process for using ROADM forexchanging data signals;

FIG. 111 illustrates the concept of a ROADM for data channels carried onmultiplexed OAM beams;

FIG. 112 illustrates observed intensity profiles at each step of anad/drop operation such as that of FIG. 111;

FIG. 113 illustrates circuitry for the generation of an OAM twisted beamusing a hologram within a micro-electromechanical device;

FIG. 114 illustrates multiple holograms generated by amicro-electromechanical device;

FIG. 115 illustrates a square array of holograms on a dark background;

FIG. 116 illustrates a hexagonal array of holograms on a darkbackground;

FIG. 117 illustrates a process for multiplexing various OAM modestogether;

FIG. 118 illustrates fractional binary fork holograms;

FIG. 119 illustrates an array of square holograms with no separation ona light background and associated generated OAM mode image;

FIG. 120 illustrates an array of circular holograms separated on a lightbackground and associated generated OAM mode image;

FIG. 121 illustrates an array of square holograms with no separation ona dark background and associated generated OAM mode image;

FIG. 122 illustrates an array of circular holograms on a dark backgroundand associated generated OAM mode image;

FIG. 123 illustrates circular holograms with separation on a brightbackground and associated generated OAM mode image;

FIG. 124 illustrates circular holograms with separation on a darkbackground and associated generated OAM mode image;

FIG. 125 illustrates a hexagonal array of circular holograms on a brightbackground and associated OAM mode image;

FIG. 126 illustrates an hexagonal array of small holograms on a brightbackground and associated OAM mode image;

FIG. 127 illustrates a hexagonal array of circular holograms on a darkbackground and associated OAM mode image;

FIG. 128 illustrates a hexagonal array of small holograms on a darkbackground and associated OAM mode image;

FIG. 129 illustrates a hexagonal array of small holograms separated on adark background and associated OAM mode image;

FIG. 130 illustrates a hexagonal array of small holograms closelylocated on a dark background and associated OAM mode image;

FIG. 131 illustrates a hexagonal array of small holograms that areseparated on a bright background and associated OAM mode image;

FIG. 132 illustrates a hexagonal array of small holograms that areclosely located on a bright background and associated OAM mode image;

FIG. 133 illustrates reduced binary holograms having a radius equal to100 micro-mirrors and a period of 50 for various OAM modes;

FIG. 134 illustrates OAM modes for holograms having a radius of 50micro-mirrors and a period of 50;

FIG. 135 illustrates OAM modes for holograms having a radius of 100micro-mirrors and a period of 100;

FIG. 136 illustrates OAM modes for holograms having a radius of 50micro-mirrors and a period of 50;

FIG. 137 illustrates additional methods of multimode OAM generation byimplementing multiple holograms within a MEMs device;

FIG. 138 illustrates binary spiral holograms;

FIG. 139 is a block diagram of a circuit for generating a muxed andmultiplexed data stream containing multiple new Eigen channels;

FIG. 140 is a flow diagram describing the operation of the circuit ofFIG. 139;

FIG. 141 is a block diagram of a circuit for de-muxing andde-multiplexing a data stream containing multiple new Eigen channels;

FIG. 142 is a flow diagram describing the operation of the circuit ofFIG. 141;

FIG. 143 illustrates various types of orthogonal functions that may beused in optical fiber transmissions;

FIG. 144 illustrates a transmitter and receiver transmitting signalsover the elliptical fiber;

FIG. 145 illustrates the process for generating an Ince-Gaussian signalfor transmission on elliptical fiber;

FIG. 146 illustrates a elliptical-cylindrical coordinate system;

FIG. 147 illustrates the curves of constant value of trace confocalellipses;

FIG. 148 illustrates a confocal hyperbola with a constant value of

FIGS. 149A and 149B illustrate the frequency of even Ince-Polynomials;

FIG. 150 illustrates the modes and phases for even Ince-Polynomials;

FIGS. 151A and 151B illustrate the frequency of odd Ince-Polynomials;

FIG. 152 illustrates the modes and phases of odd Ince-Polynomials;

FIG. 153 illustrates a parabolic index profile of an elliptical corefiber;

FIG. 154 illustrates a few mode fiber with an elliptical core;

FIG. 155 illustrates intensity diagrams for different types of beamtopologies within an elliptical core fiber;

FIG. 156 illustrates a measurement technique for generating a modecrosstalk matrix;

FIG. 157 illustrates a flow diagram of the process of FIG. 156;

FIG. 158 illustrates a generated single row of a mode crosstalk matrix;

FIG. 159 illustrates the results for a comparison of selectively excitedmodes calculated from a transmitting SLM; and

FIG. 160 illustrates a mode crosstalk matrix populated usingHermite-Gaussian modes; and

FIGS. 161 and 162 illustrate mode cross talk matrices.

DETAILED DESCRIPTION

Referring now to the drawings, wherein like reference numbers are usedherein to designate like elements throughout, the various views andembodiments of a system and method for transmissions using ellipticalcore fibers are illustrated and described, and other possibleembodiments are described. The Fig.s are not necessarily drawn to scale,and in some instances the drawings have been exaggerated and/orsimplified in places for illustrative purposes only. One of ordinaryskill in the art will appreciate the many possible applications andvariations based on the following examples of possible embodiments.

Achieving higher data capacity is perhaps one of the primary interest ofthe communications community. This is led to the investigation of usingdifferent physical properties of a light wave for communications,including amplitude, phase, wavelength and polarization. Orthogonalmodes in spatial positions are also under investigation and seemed to beuseful as well. Generally these investigative efforts can be summarizedin 2 categories: 1) encoding and decoding more bets on a single opticalpulse; a typical example is the use of advanced modulation formats,which encode information on amplitude, phase and polarization states,and 2) multiplexing and demultiplexing technologies that allow parallelpropagation of multiple independent data channels, each of which isaddressed by different light property (e.g., wavelength, polarizationand space, corresponding to wavelength-division multiplexing (WDM),polarization-division multiplexing (PDM) and space division multiplexing(SDM), respectively) using Hermite Gaussian, Laguerre Gaussian and InceGaussian spactial orthogonal modes among others.

The recognition that orbital angular momentum (OAM) has applications incommunication has made it an interesting research topic. It iswell-known that a photon can carry both spin angular momentum andorbital angular momentum. Contrary to spin angular momentum (e.g.,circularly polarized light), which is identified by the electrical fielderection, OAM is usually carried by a light beam with a helical phasefront. Due to the helical phase structure, an OAM carrying beam usuallyhas an annular intensity profile with a phase singularity at the beamcenter. Importantly, depending on discrete twisting speed of the helicalphase, OAM beams can be quantified is different states, which arecompletely distinguishable while propagating coaxially. This propertyallows OAM beams to be potentially useful in either of the 2aforementioned categories to help improve the performance of a freespace or fiber communication system. Specifically, OAM states could beused as a different dimension to encode bits on a single pulse (or asingle photon), or be used to create additional data carriers in an SDMsystem.

There are some potential benefits of using OAM for communications, somespecially designed novel fibers allow less mode coupling and cross talkwhile propagating in fibers. In addition, OAM beams with differentstates share a ring-shaped beam profile, which indicate rotationalinsensitivity for receiving the beams. Since the distinction of OAMbeams does not rely on the wavelength or polarization, OAM multiplexingcould be used in addition to WDM and PDM techniques so that potentiallyimprove the system performance may be provided.

Referring now to the drawings, and more particularly to FIG. 1, whereinthere is illustrated two manners for increasing spectral efficiency of acommunications system. In general, there are basically two ways toincrease spectral efficiency 102 of a communications system. Theincrease may be brought about by signal processing techniques 104 in themodulation scheme or using multiple access technique. Additionally, thespectral efficiency can be increase by creating new Eigen channels 106within the electromagnetic propagation. These two techniques arecompletely independent of one another and innovations from one class canbe added to innovations from the second class. Therefore, thecombination of this technique introduced a further innovation.

Spectral efficiency 102 is the key driver of the business model of acommunications system. The spectral efficiency is defined in units ofbit/sec/hz and the higher the spectral efficiency, the better thebusiness model. This is because spectral efficiency can translate to agreater number of users, higher throughput, higher quality or some ofeach within a communications system.

Regarding techniques using signal processing techniques or multipleaccess techniques. These techniques include innovations such as TDMA,FDMA, CDMA, EVDO, GSM, WCDMA, HSPA and the most recent OFDM techniquesused in 4G WIMAX and LTE. Almost all of these techniques use decades-oldmodulation techniques based on sinusoidal Eigen functions called QAMmodulation. Within the second class of techniques involving the creationof new Eigen channels 106, the innovations include diversity techniquesincluding space and polarization diversity as well as multipleinput/multiple output (MIMO) where uncorrelated radio paths createindependent Eigen channels and propagation of electromagnetic waves.

Referring now to FIG. 2, the present communication system configurationintroduces two techniques, one from the signal processing techniques 104category and one from the creation of new eigen channels 106 categorythat are entirely independent from each other. Their combinationprovides a unique manner to disrupt the access part of an end to endcommunications system from twisted pair and cable to fiber optics, tofree space optics, to RF used in cellular, backhaul and satellite, to RFsatellite, to RF broadcast, to RF point-to point, to RFpoint-to-multipoint, to RF point-to-point (backhaul), to RFpoint-to-point (fronthaul to provide higher throughput CPRI interfacefor cloudification and virtualization of RAN and cloudified HetNet), toInternet of Things (IOT), to Wi-Fi, to Bluetooth, to a personal devicecable replacement, to an RF and FSO hybrid system, to Radar, toelectromagnetic tags and to all types of wireless access. The firsttechnique involves the use of a new signal processing technique usingnew orthogonal signals to upgrade QAM modulation using non sinusoidalfunctions. This is referred to as quantum level overlay (QLO) 202. Thesecond technique involves the application of new electromagneticwavefronts using a property of electromagnetic waves or photon, calledorbital angular momentum (QAM) 104. Application of each of the quantumlevel overlay techniques 202 and orbital angular momentum application204 uniquely offers orders of magnitude higher spectral efficiency 206within communication systems in their combination.

With respect to the quantum level overlay technique 202, new eigenfunctions are introduced that when overlapped (on top of one anotherwithin a symbol) significantly increases the spectral efficiency of thesystem. The quantum level overlay technique 302 borrows from quantummechanics, special orthogonal signals that reduce the time bandwidthproduct and thereby increase the spectral efficiency of the channel.Each orthogonal signal is overlaid within the symbol acts as anindependent channel. These independent channels differentiate thetechnique from existing modulation techniques.

With respect to the application of orbital angular momentum 204, thistechnique introduces twisted electromagnetic waves, or light beams,having helical wave fronts that carry orbital angular momentum (OAM).Different OAM carrying waves/beams can be mutually orthogonal to eachother within the spatial domain, allowing the waves/beams to beefficiently multiplexed and demultiplexed within a communications link.OAM beams are interesting in communications due to their potentialability in special multiplexing multiple independent data carryingchannels.

With respect to the combination of quantum level overlay techniques 202and orbital angular momentum application 204, the combination is uniqueas the OAM multiplexing technique is compatible with otherelectromagnetic techniques such as wave length and polarization divisionmultiplexing. This suggests the possibility of further increasing systemperformance. The application of these techniques together in highcapacity data transmission disrupts the access part of an end to endcommunications system from twisted pair and cable to fiber optics, tofree space optics, to RF used in cellular, backhaul and satellite, to RFsatellite, to RF broadcast, to RF point-to point, to RFpoint-to-multipoint, to RF point-to-point (backhaul), to RFpoint-to-point (fronthaul to provide higher throughput CPRI interfacefor cloudification and virtualization of RAN and cloudified HetNet), toInternet of Things (TOT), to Wi-Fi, to Bluetooth, to a personal devicecable replacement, to an RF and FSO hybrid system, to Radar, toelectromagnetic tags and to all types of wireless access.

Each of these techniques can be applied independent of one another, butthe combination provides a unique opportunity to not only increasespectral efficiency, but to increase spectral efficiency withoutsacrificing distance or signal to noise ratios.

Using the Shannon Capacity Equation, a determination may be made ifspectral efficiency is increased. This can be mathematically translatedto more bandwidth. Since bandwidth has a value, one can easily convertspectral efficiency gains to financial gains for the business impact ofusing higher spectral efficiency. Also, when sophisticated forward errorcorrection (FEC) techniques are used, the net impact is higher qualitybut with the sacrifice of some bandwidth. However, if one can achievehigher spectral efficiency (or more virtual bandwidth), one cansacrifice some of the gained bandwidth for FEC and therefore higherspectral efficiency can also translate to higher quality.

Telecom operators and vendors are interested in increasing spectralefficiency. However, the issue with respect to this increase is thecost. Each technique at different layers of the protocol has a differentprice tag associated therewith. Techniques that are implemented at aphysical layer have the most impact as other techniques can besuperimposed on top of the lower layer techniques and thus increase thespectral efficiency further. The price tag for some of the techniquescan be drastic when one considers other associated costs. For example,the multiple input multiple output (MIMO) technique uses additionalantennas to create additional paths where each RF path can be treated asan independent channel and thus increase the aggregate spectralefficiency. In the MIMO scenario, the operator has other associated softcosts dealing with structural issues such as antenna installations, etc.These techniques not only have tremendous cost, but they have hugetiming issues as the structural activities take time and the achievingof higher spectral efficiency comes with significant delays which canalso be translated to financial losses.

The quantum level overlay technique 202 has an advantage that theindependent channels are created within the symbols without needing newantennas. This will have a tremendous cost and time benefit compared toother techniques. Also, the quantum layer overlay technique 202 is aphysical layer technique, which means there are other techniques athigher layers of the protocol that can all ride on top of the QLOtechniques 202 and thus increase the spectral efficiency even further.QLO technique 202 uses standard QAM modulation used in OFDM basedmultiple access technologies such as WIMAX or LTE. QLO technique 202basically enhances the QAM modulation at the transceiver by injectingnew signals to the I & Q components of the baseband and overlaying thembefore QAM modulation as will be more fully described herein below. Atthe receiver, the reverse procedure is used to separate the overlaidsignal and the net effect is a pulse shaping that allows betterlocalization of the spectrum compared to standard QAM or even the rootraised cosine. The impact of this technique is a significantly higherspectral efficiency.

Referring now more particularly to FIG. 3, there is illustrated ageneral overview of the manner for providing improved communicationbandwidth within various communication protocol interfaces 302, using acombination of multiple level overlay modulation 304 and the applicationof orbital angular momentum 306 to increase the number of communicationschannels.

The various communication protocol interfaces 302 may comprise a varietyof communication links, such as RF communication, wireline communicationsuch as cable or twisted pair connections, or optical communicationsmaking use of light wavelengths such as fiber-optic communications orfree-space optics. Various types of RF communications may include acombination of RF microwave or RF satellite communication, as well asmultiplexing between RF and free-space optics in real time.

By combining a multiple layer overlay modulation technique 304 withorbital angular momentum (OAM) technique 306, a higher throughput overvarious types of communication links 302 may be achieved. The use ofmultiple level overlay modulation alone without OAM increases thespectral efficiency of communication links 302, whether wired, optical,or wireless. However, with OAM, the increase in spectral efficiency iseven more significant.

Multiple overlay modulation techniques 304 provide a new degree offreedom beyond the conventional 2 degrees of freedom, with time T andfrequency F being independent variables in a two-dimensional notationalspace defining orthogonal axes in an information diagram. This comprisesa more general approach rather than modeling signals as fixed in eitherthe frequency or time domain. Previous modeling methods using fixed timeor fixed frequency are considered to be more limiting cases of thegeneral approach of using multiple level overlay modulation 304. Withinthe multiple level overlay modulation technique 304, signals may bedifferentiated in two-dimensional space rather than along a single axis.Thus, the information-carrying capacity of a communications channel maybe determined by a number of signals which occupy different time andfrequency coordinates and may be differentiated in a notationaltwo-dimensional space.

Within the notational two-dimensional space, minimization of the timebandwidth product, i.e., the area occupied by a signal in that space,enables denser packing, and thus, the use of more signals, with higherresulting information-carrying capacity, within an allocated channel.Given the frequency channel delta (Δf), a given signal transmittedthrough it in minimum time Δt will have an envelope described by certaintime-bandwidth minimizing signals. The time-bandwidth products for thesesignals take the form:

Δt Δf=1/2(2n+1)

where n is an integer ranging from 0 to infinity, denoting the order ofthe signal.

These signals form an orthogonal set of infinite elements, where eachhas a finite amount of energy. They are finite in both the time domainand the frequency domain, and can be detected from a mix of othersignals and noise through correlation, for example, by match filtering.Unlike other wavelets, these orthogonal signals have similar time andfrequency forms.

The orbital angular momentum process 306 provides a twist to wave frontsof the electromagnetic fields carrying the data stream that may enablethe transmission of multiple data streams on the same frequency,wavelength, or other signal-supporting mechanism. Similarly, otherorthogonal signals may be applied to the different data streams toenable transmission of multiple data streams on the same frequency,wavelength or other signal-supporting mechanism. This will increase thebandwidth over a communications link by allowing a single frequency orwavelength to support multiple eigen channels, each of the individualchannels having a different orthogonal and independent orbital angularmomentum associated therewith.

Referring now to FIG. 4, there is illustrated a further communicationimplementation technique using the above described techniques as twistedpairs or cables carry electrons (not photons). Rather than using each ofthe multiple level overlay modulation 304 and orbital angular momentumtechniques 306, only the multiple level overlay modulation 304 can beused in conjunction with a single wireline interface and, moreparticularly, a twisted pair communication link or a cable communicationlink 402. The operation of the multiple level overlay modulation 404, issimilar to that discussed previously with respect to FIG. 3, but is usedby itself without the use of orbital angular momentum techniques 306,and is used with either a twisted pair communication link or cableinterface communication link 402 or with fiber optics, free spaceoptics, RF used in cellular, backhaul and satellite, RF satellite, RFbroadcast, RF point-to point, RF point-to-multipoint, RF point-to-point(backhaul), RF point-to-point (fronthaul to provide higher throughputCPRI interface for cloudification and virtualization of RAN andcloudified HetNet), Internet of Things (IOT), Wi-Fi, Bluetooth, apersonal device cable replacement, an RF and FSO hybrid system, Radar,electromagnetic tags and all types of wireless access.

Referring now to FIG. 5, there is illustrated a general block diagramfor processing a plurality of data streams 502 for transmission in anoptical communication system. The multiple data streams 502 are providedto the multi-layer overlay modulation circuitry 504 wherein the signalsare modulated using the multi-layer overlay modulation technique. Themodulated signals are provided to orbital angular momentum processingcircuitry 506 which applies a twist to each of the wave fronts beingtransmitted on the wavelengths of the optical communication channel. Thetwisted waves are transmitted through the optical interface 508 over anoptical or other communications link such as an optical fiber or freespace optics communication system. FIG. 5 may also illustrate an RFmechanism wherein the interface 508 would comprise and RF interfacerather than an optical interface.

Referring now more particularly to FIG. 6, there is illustrated afunctional block diagram of a system for generating the orbital angularmomentum “twist” within a communication system, such as that illustratedwith respect to FIG. 3, to provide a data stream that may be combinedwith multiple other data streams for transmission upon a same wavelengthor frequency. Multiple data streams 602 are provided to the transmissionprocessing circuitry 600. Each of the data streams 602 comprises, forexample, an end to end link connection carrying a voice call or a packetconnection transmitting non-circuit switch packed data over a dataconnection. The multiple data streams 602 are processed bymodulator/demodulator circuitry 604. The modulator/demodulator circuitry604 modulates the received data stream 602 onto a wavelength orfrequency channel using a multiple level overlay modulation technique,as will be more fully described herein below. The communications linkmay comprise an optical fiber link, free-space optics link, RF microwavelink, RF satellite link, wired link (without the twist), etc.

The modulated data stream is provided to the orbital angular momentum(OAM) signal processing block 606. The orbital angular momentum signalprocessing block 606 applies in one embodiment an orbital angularmomentum to a signal. In other embodiments the processing block 606 canapply any orthogonal function to a signal being transmitted. Theseorthogonal functions can be spatial Bessel functions, Laguerre-Gaussianfunctions, Hermite-Gaussian functions or any other orthogonal function.Each of the modulated data streams from the modulator/demodulator 604are provided a different orbital angular momentum by the orbital angularmomentum electromagnetic block 606 such that each of the modulated datastreams have a unique and different orbital angular momentum associatedtherewith. Each of the modulated signals having an associated orbitalangular momentum are provided to an optical transmitter 608 thattransmits each of the modulated data streams having a unique orbitalangular momentum on a same wavelength. Each wavelength has a selectednumber of bandwidth slots B and may have its data transmissioncapability increase by a factor of the number of degrees of orbitalangular momentum ? that are provided from the OAM electromagnetic block606. The optical transmitter 608 transmitting signals at a singlewavelength could transmit B groups of information. The opticaltransmitter 608 and OAM electromagnetic block 606 may transmit l×Bgroups of information according to the configuration described herein.

In a receiving mode, the optical transmitter 608 will have a wavelengthincluding multiple signals transmitted therein having different orbitalangular momentum signals embedded therein. The optical transmitter 608forwards these signals to the OAM signal processing block 606, whichseparates each of the signals having different orbital angular momentumand provides the separated signals to the demodulator circuitry 604. Thedemodulation process extracts the data streams 602 from the modulatedsignals and provides it at the receiving end using the multiple layeroverlay demodulation technique.

Referring now to FIG. 7, there is provided a more detailed functionaldescription of the OAM signal processing block 606. Each of the inputdata streams are provided to OAM circuitry 702. Each of the OAMcircuitry 702 provides a different orbital angular momentum to thereceived data stream. The different orbital angular momentums areachieved by applying different currents for the generation of thesignals that are being transmitted to create a particular orbitalangular momentum associated therewith. The orbital angular momentumprovided by each of the OAM circuitries 702 are unique to the datastream that is provided thereto. An infinite number of orbital angularmomentums may be applied to different input data streams using manydifferent currents. Each of the separately generated data streams areprovided to a signal combiner 704, which combines/multiplexes thesignals onto a wavelength for transmission from the transmitter 706. Thecombiner 704 performs a spatial mode division multiplexing to place allof the signals upon a same carrier signal in the space domain.

Referring now to FIG. 8, there is illustrated the manner in which theOAM processing circuitry 606 may separate a received signal intomultiple data streams. The receiver 802 receives the combined OAMsignals on a single wavelength and provides this information to a signalseparator 804. The signal separator 804 separates each of the signalshaving different orbital angular momentums from the received wavelengthand provides the separated signals to OAM de-twisting circuitry 806. TheOAM de-twisting circuitry 806 removes the associated OAM twist from eachof the associated signals and provides the received modulated datastream for further processing. The signal separator 804 separates eachof the received signals that have had the orbital angular momentumremoved therefrom into individual received signals. The individuallyreceived signals are provided to the receiver 802 for demodulationusing, for example, multiple level overlay demodulation as will be morefully described herein below.

FIG. 9 illustrates in a manner in which a single wavelength orfrequency, having two quanti-spin polarizations may provide an infinitenumber of twists having various orbital angular momentums associatedtherewith. The/axis represents the various quantized orbital angularmomentum states which may be applied to a particular signal at aselected frequency or wavelength. The symbol omega (w) represents thevarious frequencies to which the signals of differing orbital angularmomentum may be applied. The top grid 902 represents the potentiallyavailable signals for a left handed signal polarization, while thebottom grid 904 is for potentially available signals having right handedpolarization.

By applying different orbital angular momentum states to a signal at aparticular frequency or wavelength, a potentially infinite number ofstates may be provided at the frequency or wavelength. Thus, the stateat the frequency Δω or wavelength 906 in both the left handedpolarization plane 902 and the right handed polarization plane 904 canprovide an infinite number of signals at different orbital angularmomentum states Δl. Blocks 908 and 910 represent a particular signalhaving an orbital angular momentum Δl at a frequency Δω or wavelength inboth the right handed polarization plane 904 and left handedpolarization plane 910, respectively. By changing to a different orbitalangular momentum within the same frequency Δω or wavelength 906,different signals may also be transmitted. Each angular momentum statecorresponds to a different determined current level for transmissionfrom the optical transmitter. By estimating the equivalent current forgenerating a particular orbital angular momentum within the opticaldomain and applying this current for transmission of the signals, thetransmission of the signal may be achieved at a desired orbital angularmomentum state.

Thus, the illustration of FIG. 9, illustrates two possible angularmomentums, the spin angular momentum, and the orbital angular momentum.The spin version is manifested within the polarizations of macroscopicelectromagnetism, and has only left and right hand polarizations due toup and down spin directions. However, the orbital angular momentumindicates an infinite number of states that are quantized. The paths aremore than two and can theoretically be infinite through the quantizedorbital angular momentum levels.

It is well-known that the concept of linear momentum is usuallyassociated with objects moving in a straight line. The object could alsocarry angular momentum if it has a rotational motion, such as spinning(i.e., spin angular momentum (SAM) 1002), or orbiting around an axis1006 (i.e., OAM 1004), as shown in FIGS. 10A and 10B, respectively. Alight beam may also have rotational motion as it propagates. In paraxialapproximation, a light beam carries SAM 1002 if the electrical fieldrotates along the beam axis 1006 (i.e., circularly polarized light1005), and carries OAM 1004 if the wave vector spirals around the beamaxis 1006, leading to a helical phase front 1008, as shown in FIGS. 10Cand 10D. In its analytical expression, this helical phase front 1008 isusually related to a phase term of exp(ilθ) in the transverse plane,where θ refers to the angular coordinate, and l is an integer indicatingthe number of intertwined helices (i.e., the number of 2π phase shiftsalong the circle around the beam axis). l could be a positive, negativeinteger or zero, corresponding to clockwise, counterclockwise phasehelices or a Gaussian beam with no helix, respectively.

Two important concepts relating to OAM include: 1) OAM and polarization:As mentioned above, an OAM beam is manifested as a beam with a helicalphase front and therefore a twisting wavevector, while polarizationstates can only be connected to SAM 1002. A light beam carries SAM 1002of ±h/2π (h is Plank's constant) per photon if it is left or rightcircularly polarized, and carries no SAM 1002 if it is linearlypolarized. Although the SAM 1002 and OAM 1004 of light can be coupled toeach other under certain scenarios, they can be clearly distinguishedfor a paraxial light beam. Therefore, with the paraxial assumption, OAM1004 and polarization can be considered as two independent properties oflight.

2) OAM beam and Laguerre-Gaussian (LG) beam: In general, an OAM-carryingbeam could refer to any helically phased light beam, irrespective of itsradial distribution (although sometimes OAM could also be carried by anon-helically phased beam). LG beam is a special subset among allOAM-carrying beams, due to that the analytical expression of LG beamsare eigen-solutions of paraxial form of the wave equation in acylindrical coordinates. For an LG beam, both azimuthal and radialwavefront distributions are well defined, and are indicated by two indexnumbers, l and p, of which l has the same meaning as that of a generalOAM beam, and p refers to the radial nodes in the intensitydistribution. Mathematical expressions of LG beams form an orthogonaland complete basis in the spatial domain. In contrast, a general OAMbeam actually comprises a group of LG beams (each with the same l indexbut a different p index) due to the absence of radial definition. Theterm of “OAM beam” refers to all helically phased beams, and is used todistinguish from LG beams.

Using the orbital angular momentum state of the transmitted energysignals, physical information can be embedded within the radiationtransmitted by the signals. The Maxwell-Heaviside equations can berepresented as:

${\nabla{\cdot E}} = \frac{\rho}{ɛ_{0}}$${\nabla{\times E}} = {- \frac{\partial B}{\partial t}}$ ∇⋅B = 0${\nabla{\times B}} = {{ɛ_{0}\mu_{0}\frac{\partial E}{\partial t}} + {\mu_{0}{j\left( {t,x} \right)}}}$

where ∇ is the del operator, E is the electric field intensity and B isthe magnetic flux density. Using these equations, one can derive 23symmetries/conserved quantities from Maxwell's original equations.However, there are only ten well-known conserved quantities and only afew of these are commercially used. Historically if Maxwell's equationswhere kept in their original quaternion forms, it would have been easierto see the symmetries/conserved quantities, but when they were modifiedto their present vectorial form by Heaviside, it became more difficultto see such inherent symmetries in Maxwell's equations.

Maxwell's linear theory is of U(1) symmetry with Abelian commutationrelations. They can be extended to higher symmetry group SU(2) form withnon-Abelian commutation relations that address global (non-local inspace) properties. The Wu-Yang and Harmuth interpretation of Maxwell'stheory implicates the existence of magnetic monopoles and magneticcharges. As far as the classical fields are concerned, these theoreticalconstructs are pseudo-particle, or instanton. The interpretation ofMaxwell's work actually departs in a significant ways from Maxwell'soriginal intention. In Maxwell's original formulation, Faraday'selectrotonic states (the Aμ field) was central making them compatiblewith Yang-Mills theory (prior to Heaviside). The mathematical dynamicentities called solitons can be either classical or quantum, linear ornon-linear and describe EM waves. However, solitons are of SU(2)symmetry forms. In order for conventional interpreted classicalMaxwell's theory of U(1) symmetry to describe such entities, the theorymust be extended to SU(2) forms.

Besides the half dozen physical phenomena (that cannot be explained withconventional Maxwell's theory), the recently formulated Harmuth Ansatzalso address the incompleteness of Maxwell's theory. Harmuth amendedMaxwell's equations can be used to calculate EM signal velocitiesprovided that a magnetic current density and magnetic charge are addedwhich is consistent to Yang-Mills filed equations. Therefore, with thecorrect geometry and topology, the Aμ potentials always have physicalmeaning

The conserved quantities and the electromagnetic field can berepresented according to the conservation of system energy and theconservation of system linear momentum. Time symmetry, i.e. theconservation of system energy can be represented using Poynting'stheorem according to the equations:

$H = {{\sum\limits_{i}\; {m_{i}\gamma_{i}c^{2}}} + {\frac{ɛ_{0}}{2}{\int{d^{3}{x\left( {{E}^{2} + {c^{2}{B}^{2}}} \right)}}}}}$Hamiltonian  (total  energy)${\frac{{dU}^{mech}}{dt} + \frac{{dU}^{em}}{dt} + {\oint_{s^{\prime}}{d^{2}x^{\prime}{{\hat{n}}^{\prime} \cdot S}}}} = 0$conservation  of  energy

The space symmetry, i.e., the conservation of system linear momentumrepresenting the electromagnetic Doppler shift can be represented by theequations:

$p = {{\sum\limits_{i}\; {m_{i}\gamma_{i}v_{i}}} + {ɛ_{0}{\int{d^{3}{x\left( {E \times B} \right)}}}}}$linear  momentum${\frac{{dp}^{mech}}{dt} + \frac{{dp}^{em}}{dt} + {\oint_{s^{\prime}}{d^{2}x^{\prime}{{\hat{n}}^{\prime} \cdot T}}}} = 0$conservation  of  linear  momentum

The conservation of system center of energy is represented by theequation:

$R = {{\frac{1}{H}{\sum\limits_{i}\; {\left( {x_{i} - x_{0}} \right)m_{i}\gamma_{i}c^{2}}}} + {\frac{ɛ_{0}}{2\; H}{\int{d^{3}{x\left( {x - x_{0}} \right)}\left( {{E}^{2} + {c^{2}{B}^{2}}} \right)}}}}$

Similarly, the conservation of system angular momentum, which gives riseto the azimuthal Doppler shift is represented by the equation:

${\frac{{dJ}^{mech}}{dt} + \frac{{dJ}^{em}}{dt} + {\oint_{s^{\prime}}{d^{2}x^{\prime}{{\hat{n}}^{\prime} \cdot M}}}} = 0$conservation  of  angular  momentum

For radiation beams in free space, the EM field angular momentum J^(em)can be separated into two parts:

J ^(em)=ε₀∫_(V′) d ³ x′(E×A)+ε₀∫_(V′) d ₃ x′E _(i)[(x′−x ₀)×∇]A _(i)

For each singular Fourier mode in real valued representation:

$J^{em} = {{{- i}\frac{ɛ_{0}}{2\omega}{\int_{V^{\prime}}{d^{3}{x^{\prime}\left( {E^{*} \times E} \right)}}}} - {i\frac{ɛ_{0}}{2\omega}{\int_{V^{\prime}}{d^{3}x^{\prime}{E_{i}\left\lbrack {\left( {x^{\prime} - x_{0}} \right) \times \nabla} \right\rbrack}E_{i}}}}}$

The first part is the EM spin angular momentum S^(em), its classicalmanifestation is wave polarization. And the second part is the EMorbital angular momentum L^(em) its classical manifestation is wavehelicity. In general, both EM linear momentum P^(em), and EM angularmomentum J^(em)=L^(em)+S^(em) are radiated all the way to the far field.

By using Poynting theorem, the optical vorticity of the signals may bedetermined according to the optical velocity equation:

${{\frac{\partial U}{\partial t} + {\nabla{\cdot S}}} = 0},$

continuity equationwhere S is the Poynting vector

S=¼(E×H*+E*×H),

and U is the energy density

U=¼(ε|E| ²+μ₀ |H| ₂),

with E and H comprising the electric field and the magnetic field,respectively, and ε and μ₀ being the permittivity and the permeabilityof the medium, respectively. The optical vorticity V may then bedetermined by the curl of the optical velocity according to theequation:

$V = {{\nabla{\times v_{opt}}} = {\nabla{\times \left( \frac{{E \times H^{*}} + {E^{*} \times H}}{{ɛ{E}^{2}} + {\mu_{0}{H}^{2}}} \right)}}}$

Referring now to FIGS. 10A and 10B, there is illustrated the manner inwhich a signal and its associated Poynting vector in a plane wavesituation. In the plane wave situation illustrated generally at 1002,the transmitted signal may take one of three configurations. When theelectric field vectors are in the same direction, a linear signal isprovided, as illustrated generally at 1004. Within a circularpolarization 1006, the electric field vectors rotate with the samemagnitude. Within the elliptical polarization 1008, the electric fieldvectors rotate but have differing magnitudes. The Poynting vectorremains in a constant direction for the signal configuration to FIG. 10Aand always perpendicular to the electric and magnetic fields. Referringnow to FIG. 10B, when a unique orbital angular momentum is applied to asignal as described here and above, the Poynting vector S 1010 willspiral about the direction of propagation of the signal. This spiral maybe varied in order to enable signals to be transmitted on the samefrequency as described herein.

FIGS. 11A through 11C illustrate the differences in signals havingdifferent helicity (i.e., orbital angular momentums). Each of thespiraling Poynting vectors associated with the signals 1102, 1104, and1106 provide a different shaped signal. Signal 1102 has an orbitalangular momentum of +1, signal 1104 has an orbital angular momentum of+3, and signal 1106 has an orbital angular momentum of −4. Each signalhas a distinct angular momentum and associated Poynting vector enablingthe signal to be distinguished from other signals within a samefrequency. This allows differing type of information to be transmittedon the same frequency, since these signals are separately detectable anddo not interfere with each other (Eigen channels).

FIG. 11D illustrates the propagation of Poynting vectors for variousEigen modes. Each of the rings 1120 represents a different Eigen mode ortwist representing a different orbital angular momentum within the samefrequency. Each of these rings 1120 represents a different orthogonalchannel. Each of the Eigen modes has a Poynting vector 1122 associatedtherewith.

Topological charge may be multiplexed to the frequency for either linearor circular polarization. In case of linear polarizations, topologicalcharge would be multiplexed on vertical and horizontal polarization. Incase of circular polarization, topological charge would multiplex onleft hand and right hand circular polarizations. The topological chargeis another name for the helicity index “I” or the amount of twist or OAMapplied to the signal. Also, use of the orthogonal functions discussedherein above may also be multiplexed together onto a same signal inorder to transmit multiple streams of information. The helicity indexmay be positive or negative. In wireless communications, differenttopological charges/orthogonal functions can be created and muxedtogether and de-muxed to separate the topological chargescharges/orthogonal functions. The signals having different orthogonalfunction are spatially combined together on a same signal but do notinterfere with each other since they are orthogonal to each other.

The topological charges l s can be created using Spiral Phase Plates(SPPs) as shown in FIG. 11E using a proper material with specific indexof refraction and ability to machine shop or phase mask, hologramscreated of new materials or a new technique to create an RF version ofSpatial Light Modulator (SLM) that does the twist of the RF waves (asopposed to optical beams) by adjusting voltages on the device resultingin twisting of the RF waves with a specific topological charge. SpiralPhase plates can transform a RF plane wave (l=0) to a twisted RF wave ofa specific helicity (i.e. l=+1).

Cross talk and multipath interference can be corrected using RFMultiple-Input-Multiple-Output (MIMO). Most of the channel impairmentscan be detected using a control or pilot channel and be corrected usingalgorithmic techniques (closed loop control system).

While the application of orbital angular momentum to various signalsallow the signals to be orthogonal to each other and used on a samesignal carrying medium, other orthogonal function/signals can be appliedto data streams to create the orthogonal signals on the same signalmedia carrier.

Within the notational two-dimensional space, minimization of the timebandwidth product, i.e., the area occupied by a signal in that space,enables denser packing, and thus, the use of more signals, with higherresulting information-carrying capacity, within an allocated channel.Given the frequency channel delta (Δf), a given signal transmittedthrough it in minimum time Δt will have an envelope described by certaintime-bandwidth minimizing signals. The time-bandwidth products for thesesignals take the form;

Δt Δf=1/2(2n+1)

where n is an integer ranging from 0 to infinity, denoting the order ofthe signal.

These signals form an orthogonal set of infinite elements, where eachhas a finite amount of energy. They are finite in both the time domainand the frequency domain, and can be detected from a mix of othersignals and noise through correlation, for example, by match filtering.Unlike other wavelets, these orthogonal signals have similar time andfrequency forms. These types of orthogonal signals that reduce the timebandwidth product and thereby increase the spectral efficiency of thechannel.

Hermite-Gaussian polynomials are one example of a classical orthogonalpolynomial sequence, which are the Eigenstates of a quantum harmonicoscillator. Signals based on Hermite-Gaussian polynomials possess theminimal time-bandwidth product property described above, and may be usedfor embodiments of MLO systems. However, it should be understood thatother signals may also be used, for example orthogonal polynomials suchas Jacobi polynomials, Gegenbauer polynomials, Legendre polynomials,Chebyshev polynomials, and Laguerre-Gaussian polynomials. Q-functionsare another class of functions that can be employed as a basis for MLOsignals.

In addition to the time bandwidth minimization described above, theplurality of data streams can be processed to provide minimization ofthe Space-Momentum products in spatial modulation. In this case:

ΔxΔp=½

Processing of the data streams in this manner create wavefronts that arespatial. The processing creates wavefronts that are also orthogonal toeach other like the OAM twisted functions but these comprise differenttypes of orthogonal functions that are in the spatial domain rather thanthe temporal domain.

The above described scheme is applicable to twisted pair, coaxial cable,fiber optic, RF satellite, RF broadcast, RF point-to point, RFpoint-to-multipoint, RF point-to-point (backhaul), RF point-to-point(fronthaul to provide higher throughput CPRI interface forcloudification and virtualization of RAN and cloudified HetNet),free-space optics (FSO), Internet of Things (IOT), Wifi, Bluetooth, as apersonal device cable replacement, RF and FSO hybrid system, Radar,electromagnetic tags and all types of wireless access. The method andsystem are compatible with many current and future multiple accesssystems, including EV-DO, UMB, WIMAX, WCDMA (with or without),multimedia broadcast multicast service (MBMS)/multiple input multipleoutput (MIMO), HSPA evolution, and LTE.

Hermite Gaussian Beams

Hermite Gaussian beams may also be used for transmitting orthogonal datastreams. In the scalar field approximation (e.g. neglecting the vectorcharacter of the electromagnetic field), any electric field amplitudedistribution can be represented as a superposition of plane waves, i.e.by:

$E \propto {\int{\int{\frac{{dk}_{x}{dk}_{y}}{\left( {2\pi} \right)^{2}}{A\left( {k_{x},k_{y}} \right)}e^{{{ik}_{x}x} + {{ik}_{y}y} + {{ik}_{z}z} + {{iz}\sqrt{k^{2} - k_{x}^{2} - k_{y}^{2}}}}}}}$

This representation is also called angular spectrum of plane waves orplane-wave expansion of the electromagnetic field. Here A(k_(x), k_(y))is the amplitude of the plane wave. This representation is chosen insuch a way that the net energy flux connected with the electromagneticfield is towards the propagation axis z. Every plane wave is connectedwith an energy flow that has direction k. Actual lasers generate aspatially coherent electromagnetic field which has a finite transversalextension and propagates with moderate spreading. That means that thewave amplitude changes only slowly along the propagation axis (z-axis)compared to the wavelength and finite width of the beam. Thus, theparaxial approximation can be applied, assuming that the amplitudefunction A(k_(x), k_(y)) falls off sufficiently fast with increasingvalues of (k_(x), k_(y)).

Two principal characteristics of the total energy flux can beconsidered: the divergence (spread of the plane wave amplitudes in wavevector space), defined as:

${Divergence} \propto {\int{\int{\frac{{dk}_{x}{dk}_{y}}{\left( {2\pi} \right)^{2}}\left( {K_{x}^{2} + K_{y}^{2}} \right){{A\left( {k_{x},k_{y}} \right)}}^{2}}}}$

and the transversal spatial extension (spread of the field intensityperpendicular to the z-direction) defined as:

${{{Transversal}\mspace{14mu} {Extention}} \propto {\int_{- \infty}^{\infty}{{dx}{\int_{- \infty}^{\infty}{{dy}\; \left( {x^{2} + y^{2}} \right){E}^{2}}}}}} = {\int{\int{\frac{{dk}_{x}{dk}_{y}}{\left( {2\pi} \right)^{2}}\left\lbrack {{\frac{\partial A}{\partial x}}^{2} + {\frac{\partial A}{\partial y}}^{2}} \right\rbrack}}}$

Let's now look for the fundamental mode of the beam as theelectromagnetic field having simultaneously minimal divergence andminimal transversal extension, i.e. as the field that minimizes theproduct of divergence and extension. By symmetry reasons, this leads tolooking for an amplitude function minimizing the product:

${\left\lbrack {\int_{- \infty}^{\infty}{\frac{{dk}_{x}}{\left( {2\pi} \right)}k_{x}^{2}{A}^{2}}} \right\rbrack \left\lbrack {\int_{- \infty}^{\infty}{\frac{{dk}_{x}}{\left( {2\pi} \right)}{\frac{\partial A}{\partial k_{x}}}^{2}}} \right\rbrack} = \frac{{A}^{4}}{\left( {8\pi^{2}} \right)^{2}}$

Thus, seeking the field with minimal divergence and minimal transversalextension can lead directly to the fundamental Gaussian beam. This meansthat the Gaussian beam is the mode with minimum uncertainty, i.e. theproduct of its sizes in real space and wave-vector space is thetheoretical minimum as given by the Heisenberg's uncertainty principleof Quantum Mechanics. Consequently, the Gaussian mode has lessdispersion than any other optical field of the same size, and itsdiffraction sets a lower threshold for the diffraction of real opticalbeams.

Hermite-Gaussian beams are a family of structurally stable laser modeswhich have rectangular symmetry along the propagation axis. In order toderive such modes, the simplest approach is to include an additionalmodulation of the form:

$E_{m,n}^{H} = {\int_{- \infty}^{\infty}{\frac{{dk}_{x}{dk}_{y}}{\left( {2\pi} \right)^{2}}\left( {ik}_{x} \right)^{m}\left( {ik}_{y} \right)^{n}e^{s}}}$${S\left( {k_{x},k_{y},x,y,z} \right)} = {{{ik}_{y}x} + {{ik}_{y}y} + {{ik}_{z}z} - {\frac{W_{0}}{4}{\left( {1 + {i\frac{Z}{Z_{R}}}} \right)\left\lbrack {k_{x}^{2} + k_{y}^{2}} \right\rbrack}}}$

The new field modes occur to be differential derivatives of thefundamental Gaussian mode E₀.

$E_{m,n}^{H} = {\frac{\partial^{m + n}}{{\partial x^{m}}{\partial y^{n}}}E_{0}}$

Looking at the explicit form E0 shows that the differentiations in thelast equation lead to expressions of the form:

$\frac{\partial^{P}}{\partial x^{p}}e^{({{- \alpha}\; x^{2}})}$

with some constant p and α. Using now the definition of Hermits'polynomials,

${H_{p}(x)} = {\left( {- 1} \right)^{p}e^{(x^{2})}\frac{d^{P}}{{dx}^{p}}e^{({{- \alpha}\; x^{2}})}}$

Then the field amplitude becomes

${E_{m,n}^{H}\left( {x,y,z} \right)} = {\sum\limits_{m}\; {\sum\limits_{n}\; {C_{mn}E_{0}\frac{w_{0}}{w(z)}{H_{m}\left( {\sqrt{2}\frac{x}{w(z)}} \right)}{H_{n}\left( {\sqrt{2}\frac{y}{w(z)}} \right)}e^{\frac{- {({x^{2} + y^{2}})}}{{w{(z)}}^{2}}}e^{{- {j{({m + n + 1})}}}\tan^{- 1}{z/z_{R}}}e^{\frac{- {({x^{2} + y^{2}})}}{2\; {R{(z)}}}}}}}$

Where

ρ² =x ² +y ²

ξ=z/z _(R)

and Rayleigh length z_(R)

$z_{R} = \frac{\pi \; w_{0}^{2}}{\lambda}$

And beam diameter

w(ξ)=w ₀√{square root over ((1+ξ²))}

In cylindrical coordinates, the filed takes the form:

${E_{l,p}^{L}\left( {\rho,\phi,z} \right)} = {\sum\limits_{l}\; {\sum\limits_{np}\; {C_{lp}E_{0}\frac{w_{0}}{w(z)}\left( {\sqrt{2}\frac{\rho}{w(z)}} \right)^{l}{L_{p}^{l}\left( {\sqrt{2}\frac{\rho}{w(z)}} \right)}e^{\frac{- \rho^{2}}{{w{(z)}}^{2}}}e^{{- {j{({{2\; p} + l + 1})}}}\tan^{- 1}{z/z_{R}}}e^{{jl}\; \phi}e^{\frac{{- {jk}}\; \rho^{2}}{2\; {R{(z)}}}}}}}$

Where L_(p) ^(l) is Laguerre functions.

Mode division multiplexing (MDM) of multiple orthogonal beams increasesthe system capacity and spectral efficiency in optical communicationsystems. For free space systems, multiple beams each on a differentorthogonal mode can be transmitted through a single transmitter andreceiver aperture pair. Moreover, the modal orthogonality of differentbeans enables the efficient multiplexing at the transmitter anddemultiplexing at the receiver.

Different optical modal basis sets exist that exhibit orthogonality. Forexample, orbital angular momentum (OAM) beams that are either LaguerreGaussian (LG or Laguerre Gaussian light modes may be used formultiplexing of multiple orthogonal beams in free space optical and RFtransmission systems. However, there exist other modal groups that alsomay be used for multiplexing that do not contain OAM. Hermite Gaussian(HG) modes are one such modal group. The intensity of an HG_(m,n) beamis shown according to the equation:

$\begin{matrix}{{{I\left( {x,y,z} \right)} = {C_{m,n}{H_{m}^{2}\left( \frac{\sqrt{2}x}{w(z)} \right)}{H_{n}^{2}\left( \frac{\sqrt{2}y}{w(z)} \right)} \times {\exp \left( {{- \frac{2\; x^{2}}{{w(z)}^{2}}} - \frac{2\; y^{2}}{{w(z)}^{2}}} \right)}}},{w(z)}} \\{= {w_{0}\sqrt{1 + \left\lbrack {\lambda \; {z/\pi}\; w_{0}^{2}} \right\rbrack}}}\end{matrix}$

in which H_(m)(*) and He) are the Hermite polynomials of the mth and nthorder. The value w₀ is the beam waist at distance Z=0. The spatialorthogonality of HG modes with the same beam waist w₀ relies on theorthogonality of Hermite polynomial in x or y directions.

Referring now to FIG. 13, there is illustrated a system for using theorthogonality of an HG modal group for free space spatial multiplexingin free space. A laser 1302 is provided to a beam splitter 1304. Thebeam splitter 1304 splits the beam into multiple beams that are eachprovided to a modulator 1306 for modulation with a data stream 1308. Themodulated beam is provided to collimators 1310 that provides acollimated light beam to spatial light modulators 1312. Spatial lightmodulators (SLM's) 1312 may be used for transforming input plane wavesinto HG modes of different orders, each mode carrying an independentdata channel. These HG modes are spatially multiplexed using amultiplexer 1314 and coaxially transmitted over a free space link 1316.At the receiver 1318 there are several factors that may affect thedemultiplexing of these HG modes, such as receiver aperture size,receiver lateral displacement and receiver angular error. These factorsaffect the performance of the data channel such as signal-to-noise ratioand crosstalk.

With respect to the characteristics of a diverged HG_(m,0) beam (m=0-6),the wavelength is assumed to be 1550 nm and the transmitted power foreach mode is 0 dBm. Higher order HG modes have been shown to have largerbeam sizes. For smaller aperture sizes less power is received for higherorder HG modes due to divergence.

Since the orthogonality of HG modes relies on the optical fielddistribution in the x and y directions, a finite receiver aperture maytruncate the beam. The truncation will destroy the orthogonality andcost crosstalk of the HG channels. When an aperture is smaller, there ishigher crosstalk to the other modes. When a finite receiver is used, ifan HG mode with an even (odd) order is transmitted, it only causes crosstalk to other HG modes with even (odd) numbers. This is explained by thefact that the orthogonality of the odd and even HG modal groups remainswhen the beam is systematically truncated.

Moreover, misalignment of the receiver may cause crosstalk. In oneexample, lateral displacement can be caused when the receiver is notaligned with the beam axis. In another example, angular error may becaused when the receiver is on axis but there is an angle between thereceiver orientation and the beam propagation axis. As the lateraldisplacement increases, less power is received from the transmittedpower mode and more power is leaked to the other modes. There is lesscrosstalk for the modes with larger mode index spacing from thetransmitted mode.

Referring now to FIG. 14, the reference number 1400 generally indicatesan embodiment of a multiple level overlay (MLO) modulation system,although it should be understood that the term MLO and the illustratedsystem 1400 are examples of embodiments. The MLO system may comprise onesuch as that disclosed in U.S. Pat. No. 8,503,546 entitled MultipleLayer Overlay Modulation which is incorporated herein by reference. Inone example, the modulation system 1400 would be implemented within themultiple level overlay modulation box 504 of FIG. 5. System 1400 takesas input an input data stream 1401 from a digital source 1402, which isseparated into three parallel, separate data streams, 1403A-1403C, oflogical 1s and 0s by input stage demultiplexer (DEMUX) 1404. Data stream1401 may represent a data file to be transferred, or an audio or videodata stream. It should be understood that a greater or lesser number ofseparated data streams may be used. In some of the embodiments, each ofthe separated data streams 1403A-1403C has a data rate of 1/N of theoriginal rate, where N is the number of parallel data streams. In theembodiment illustrated in FIG. 14, N is 3.

Each of the separated data streams 1403A-1403C is mapped to a quadratureamplitude modulation (QAM) symbol in an M-QAM constellation, forexample, 16 QAM or 64 QAM, by one of the QAM symbol mappers 1405A-C. TheQAM symbol mappers 1405A-C are coupled to respective outputs of DEMUX1404, and produced parallel in phase (I) 1406A, 1408A, and 1410A andquadrature phase (Q) 1406B, 1408B, and 1410B data streams at discretelevels. For example, in 64 QAM, each I and Q channel uses 8 discretelevels to transmit 3 bits per symbol. Each of the three I and Q pairs,1406A-1406B, 1408A-1408B, and 1410A-1410B, is used to weight the outputof the corresponding pair of function generators 1407A-1407B,1409A-1409B, and 1411A-1411B, which in some embodiments generate signalssuch as the modified Hermite polynomials described above and weightsthem based on the amplitude value of the input symbols. This provides 2Nweighted or modulated signals, each carrying a portion of the dataoriginally from income data stream 1401, and is in place of modulatingeach symbol in the I and Q pairs, 1406A-1406B, 1408A-1408B, and1410A-1410B with a raised cosine filter, as would be done for a priorart QAM system. In the illustrated embodiment, three signals are used,SH0, SH1, and SH2, which correspond to modifications of H0, H1, and H2,respectively, although it should be understood that different signalsmay be used in other embodiments.

While the description relates to the application of QLO modulation toimprove operation of a quadrature amplitude modulation (QAM) system, theapplication of QLO modulation will also improve the spectral efficiencyof other legacy modulation schemes.

The weighted signals are not subcarriers, but rather are sublayers of amodulated carrier, and are combined, superimposed in both frequency andtime, using summers 1412 and 1416, without mutual interference in eachof the I and Q dimensions, due to the signal orthogonality. Summers 1412and 1416 act as signal combiners to produce composite signals 1413 and1417. The weighted orthogonal signals are used for both I and Qchannels, which have been processed equivalently by system 1400, and aresummed before the QAM signal is transmitted. Therefore, although neworthogonal functions are used, some embodiments additionally use QAM fortransmission. Because of the tapering of the signals in the time domain,as will be shown in FIGS. 18A through 18K, the time domain waveform ofthe weighted signals will be confined to the duration of the symbols.Further, because of the tapering of the special signals and frequencydomain, the signal will also be confined to frequency domain, minimizinginterface with signals and adjacent channels.

The composite signals 1413 and 1417 are converted to analogue signals1415 and 1419 using digital to analogue converters 1414 and 1418, andare then used to modulate a carrier signal at the frequency of localoscillator (LO) 1420, using modulator 1421. Modulator 1421 comprisesmixers 1422 and 1424 coupled to DACs 1414 and 1418, respectively. Ninetydegree phase shifter 1423 converts the signals from LO 1420 into a Qcomponent of the carrier signal. The output of mixers 1422 and 1424 aresummed in summer 1425 to produce output signals 1426.

MLO can be used with a variety of transport mediums, such as wire,optical, and wireless, and may be used in conjunction with QAM. This isbecause MLO uses spectral overlay of various signals, rather thanspectral overlap. Bandwidth utilization efficiency may be increased byan order of magnitude, through extensions of available spectralresources into multiple layers. The number of orthogonal signals isincreased from 2, cosine and sine, in the prior art, to a number limitedby the accuracy and jitter limits of generators used to produce theorthogonal polynomials. In this manner, MLO extends each of the I and Qdimensions of QAM to any multiple access techniques such as GSM, codedivision multiple access (CDMA), wide band CDMA (WCDMA), high speeddownlink packet access (HSPDA), evolution-data optimized (EV-DO),orthogonal frequency division multiplexing (OFDM), world-wideinteroperability for microwave access (WIMAX), and long term evolution(LTE) systems. MLO may be further used in conjunction with othermultiple access (MA) schemes such as frequency division duplexing (FDD),time division duplexing (TDD), frequency division multiple access(FDMA), and time division multiple access (TDMA). Overlaying individualorthogonal signals over the same frequency band allows creation of avirtual bandwidth wider than the physical bandwidth, thus adding a newdimension to signal processing. This modulation is applicable to twistedpair, coaxial cable, fiber optic, RF satellite, RF broadcast, RFpoint-to point, RF point-to-multipoint, RF point-to-point (backhaul), RFpoint-to-point (fronthaul to provide higher throughput CPRI interfacefor cloudification and virtualization of RAN and cloudified HetNet),free-space optics (FSO), Internet of Things (TOT), Wifi, Bluetooth, as apersonal device cable replacement, RF and FSO hybrid system, Radar,electromagnetic tags and all types of wireless access. The method andsystem are compatible with many current and future multiple accesssystems, including EV-DO, UMB, WIMAX, WCDMA (with or without),multimedia broadcast multicast service (MBMS)/multiple input multipleoutput (MIMO), HSPA evolution, and LTE.

Referring now back to FIG. 15, an MLO demodulator 1500 is illustrated,although it should be understood that the term MLO and the illustratedsystem 1500 are examples of embodiments. The modulator 1500 takes asinput an MLO signal 1526 which may be similar to output signal 1526 fromsystem 1400. Synchronizer 1527 extracts phase information, which isinput to local oscillator 1520 to maintain coherence so that themodulator 1521 can produce base band to analogue I signal 1515 and Qsignal 1519. The modulator 1521 comprises mixers 1522 and 1524, which,coupled to OL 1520 through 90 degree phase shifter 1523. I signal 1515is input to each of signal filters 1507A, 1509A, and 1511A, and Q signal1519 is input to each of signal filters 1507B, 1509B, and 1511B. Sincethe orthogonal functions are known, they can be separated usingcorrelation or other techniques to recover the modulated data.Information in each of the I and Q signals 1515 and 1519 can beextracted from the overlapped functions which have been summed withineach of the symbols because the functions are orthogonal in acorrelative sense.

In some embodiments, signal filters 1507A-1507B, 1509A-1509B, and1511A-1511B use locally generated replicas of the polynomials as knownsignals in match filters. The outputs of the match filters are therecovered data bits, for example, equivalence of the QAM symbols1506A-1506B, 1508A-1508B, and 1510A-1510B of system 1500. Signal filters1507A-1507B, 1509A-1509B, and 1511A-1511B produce 2n streams of n, I,and Q signal pairs, which are input into demodulators 1528-1533.Demodulators 1528-1533 integrate the energy in their respective inputsignals to determine the value of the QAM symbol, and hence the logicalis and Os data bit stream segment represented by the determined symbol.The outputs of the modulators 1528-1533 are then input into multiplexers(MUXs) 1505A-1505C to generate data streams 1503A-1503C. If system 1500is demodulating a signal from system 1400, data streams 1503A-1503Ccorrespond to data streams 1403A-1403C. Data streams 1503A-1503C aremultiplexed by MUX 1504 to generate data output stream 1501. In summary,MLO signals are overlayed (stacked) on top of one another on transmitterand separated on receiver.

MLO may be differentiated from CDMA or OFDM by the manner in whichorthogonality among signals is achieved. MLO signals are mutuallyorthogonal in both time and frequency domains, and can be overlaid inthe same symbol time bandwidth product. Orthogonality is attained by thecorrelation properties, for example, by least sum of squares, of theoverlaid signals. In comparison, CDMA uses orthogonal interleaving ordisplacement of signals in the time domain, whereas OFDM uses orthogonaldisplacement of signals in the frequency domain.

Bandwidth efficiency may be increased for a channel by assigning thesame channel to multiple users. This is feasible if individual userinformation is mapped to special orthogonal functions. CDMA systemsoverlap multiple user information and views time intersymbol orthogonalcode sequences to distinguish individual users, and OFDM assigns uniquesignals to each user, but which are not overlaid, are only orthogonal inthe frequency domain. Neither CDMA nor OFDM increases bandwidthefficiency. CDMA uses more bandwidth than is necessary to transmit datawhen the signal has a low signal to noise ratio (SNR). OFDM spreads dataover many subcarriers to achieve superior performance in multipathradiofrequency environments. OFDM uses a cyclic prefix OFDM to mitigatemultipath effects and a guard time to minimize intersymbol interference(ISI), and each channel is mechanistically made to behave as if thetransmitted waveform is orthogonal. (Sync function for each subcarrierin frequency domain.)

In contrast, MLO uses a set of functions which effectively form analphabet that provides more usable channels in the same bandwidth,thereby enabling high bandwidth efficiency. Some embodiments of MLO donot require the use of cyclic prefixes or guard times, and therefore,outperforms OFDM in spectral efficiency, peak to average power ratio,power consumption, and requires fewer operations per bit. In addition,embodiments of MLO are more tolerant of amplifier nonlinearities thanare CDMA and OFDM systems.

FIG. 16 illustrates an embodiment of an MLO transmitter system 1600,which receives input data stream 1601. System 1600 represents amodulator/controller, which incorporates equivalent functionality ofDEMUX 1604, QAM symbol mappers 1405A-C, function generators 1407A-1407B,1409A-1409B, and 1411A-1411B, and summers 1412 and 1416 of system 1400,shown in FIG. 14. However, it should be understood thatmodulator/controller 1601 may use a greater or lesser quantity ofsignals than the three illustrated in system 1400. Modulator/controller1601 may comprise an application specific integrated circuit (ASIC), afield programmable gate array (FPGA), and/or other components, whetherdiscrete circuit elements or integrated into a single integrated circuit(IC) chip.

Modulator/controller 1601 is coupled to DACs 1604 and 1607,communicating a 10 bit I signal 1602 and a 10 bit Q signal 1605,respectively. In some embodiments, I signal 1602 and Q signal 1605correspond to composite signals 1413 and 1417 of system 1400. It shouldbe understood, however, that the 10 bit capacity of I signal 1602 and Qsignal 1605 is merely representative of an embodiment. As illustrated,modulator/controller 1601 also controls DACs 1604 and 1607 using controlsignals 1603 and 1606, respectively. In some embodiments, DACs 1604 and1607 each comprise an AD5433, complementary metal oxide semiconductor(CMOS) 10 bit current output DAC. In some embodiments, multiple controlsignals are sent to each of DACs 1604 and 1607.

DACs 1604 and 1607 output analogue signals 1415 and 1419 to quadraturemodulator 1421, which is coupled to LO 1420. The output of modulator1420 is illustrated as coupled to a transmitter 1608 to transmit datawirelessly, although in some embodiments, modulator 1421 may be coupledto a fiber-optic modem, a twisted pair, a coaxial cable, or othersuitable transmission media.

FIG. 15 illustrates an embodiment of an MLO receiver system 1500 capableof receiving and demodulating signals from system 1600. System 1500receives an input signal from a receiver 1508 that may comprise inputmedium, such as RF, wired or optical. The modulator 1321 driven by LO1320 converts the input to baseband I signal 1315 and Q signal 1319. Isignal 1315 and Q signal 1319 are input to analogue to digital converter(ADC) 1509.

ADC 1709 outputs 10 bit signal 1710 to demodulator/controller 1701 andreceives a control signal 1712 from demodulator/controller 1701.Demodulator/controller 1701 may comprise an application specificintegrated circuit (ASIC), a field programmable gate array (FPGA),and/or other components, whether discrete circuit elements or integratedinto a single integrated circuit (IC) chip. Demodulator/controller 1701correlates received signals with locally generated replicas of thesignal set used, in order to perform demodulation and identify thesymbols sent. Demodulator/controller 1701 also estimates frequencyerrors and recovers the data clock, which is used to read data from theADC 1709. The clock timing is sent back to ADC 1709 using control signal1712, enabling ADC 1709 to segment the digital I and Q signals 1517 and1519. In some embodiments, multiple control signals are sent bydemodulator/controller 1701 to ADC 1709. Demodulator/controller 1701also outputs data signal 1301.

Hermite-Gaussian polynomials are a classical orthogonal polynomialsequence, which are the Eigenstates of a quantum harmonic oscillator.Signals based on Hermite-Gaussian polynomials possess the minimaltime-bandwidth product property described above, and may be used forembodiments of MLO systems. However, it should be understood that othersignals may also be used, for example orthogonal polynomials such asJacobi polynomials, Gegenbauer polynomials, Legendre polynomials,Chebyshev polynomials, and Laguerre-Gaussian polynomials. Q-functionsare another class of functions that can be employed as a basis for MLOsignals.

In quantum mechanics, a coherent state is a state of a quantum harmonicoscillator whose dynamics most closely resemble the oscillating behaviorof a classical harmonic oscillator system. A squeezed coherent state isany state of the quantum mechanical Hilbert space, such that theuncertainty principle is saturated. That is, the product of thecorresponding two operators takes on its minimum value. In embodimentsof an MLO system, operators correspond to time and frequency domainswherein the time-bandwidth product of the signals is minimized. Thesqueezing property of the signals allows scaling in time and frequencydomain simultaneously, without losing mutual orthogonality among thesignals in each layer. This property enables flexible implementations ofMLO systems in various communications systems.

Because signals with different orders are mutually orthogonal, they canbe overlaid to increase the spectral efficiency of a communicationchannel. For example, when n=0, the optimal baseband signal will have atime-bandwidth product of ½, which is the Nyquist Inter-SymbolInterference (ISI) criteria for avoiding ISI. However, signals withtime-bandwidth products of 3/2, 5/2, 7/2, and higher, can be overlaid toincrease spectral efficiency.

An embodiment of an MLO system uses functions based on modified Hermitepolynomials, 4n, and are defined by:

${\psi_{n}\left( {t,\xi} \right)} = {\frac{\left( {\tanh \; \xi} \right)^{n/2}}{2^{n/2}\left( {{n!}\cosh \; \xi} \right)^{1/2}}e^{\frac{1}{2}{t^{2}{\lbrack{1 - {\tanh \; \xi}}\rbrack}}}{H_{n}\left( \frac{t}{\sqrt{2\; \cosh \; \xi \; \sinh \; \xi}} \right)}}$

where t is time, and is a bandwidth utilization parameter. Plots ofΨ_(n) for n ranging from 0 to 9, along with their Fourier transforms(amplitude squared), are shown in FIGS. 5A-5K. The orthogonality ofdifferent orders of the functions may be verified by integrating:

∫∫ψ_(n)(t,ξ)ψ_(m)(t,ξ)dtdξ

The Hermite polynomial is defined by the contour integral:

${H_{n}(z)} = {\frac{n!}{2{\pi!}}{\oint{e^{{- t^{2}} + {2\; t\; 2}}t^{{- n} - 1}{dt}}}}$

where the contour encloses the origin and is traversed in acounterclockwise direction. Hermite polynomials are described inMathematical Methods for Physicists, by George Arfken, for example onpage 416, the disclosure of which is incorporated by reference.

FIGS. 18A-18K illustrate representative MLO signals and their respectivespectral power densities based on the modified Hermite polynomials Ψ_(n)for n ranging from 0 to 9. FIG. 18A shows plots 1801 and 1804. Plot 1801comprises a curve 1827 representing Ψ₀ plotted against a time axis 1802and an amplitude axis 1803. As can be seen in plot 1801, curve 1827approximates a Gaussian curve. Plot 1804 comprises a curve 1837representing the power spectrum of Ψ₀ plotted against a frequency axis1805 and a power axis 1806. As can be seen in plot 1804, curve 1837 alsoapproximates a Gaussian curve. Frequency domain curve 1807 is generatedusing a Fourier transform of time domain curve 1827. The units of timeand frequency on axis 1802 and 1805 are normalized for basebandanalysis, although it should be understood that since the time andfrequency units are related by the Fourier transform, a desired time orfrequency span in one domain dictates the units of the correspondingcurve in the other domain. For example, various embodiments of MLOsystems may communicate using symbol rates in the megahertz (MHz) orgigahertz (GHz) ranges and the non-0 duration of a symbol represented bycurve 1827, i.e., the time period at which curve 1827 is above 0 wouldbe compressed to the appropriate length calculated using the inverse ofthe desired symbol rate. For an available bandwidth in the megahertzrange, the non-0 duration of a time domain signal will be in themicrosecond range.

FIGS. 18B-18J show plots 1807-1824, with time domain curves 1828-1836representing Ψ₁ through Ψ₉, respectively, and their correspondingfrequency domain curves 1838-1846. As can be seen in FIGS. 18A-18J, thenumber of peaks in the time domain plots, whether positive or negative,corresponds to the number of peaks in the corresponding frequency domainplot. For example, in plot 1823 of FIG. 18J, time domain curve 1836 hasfive positive and five negative peaks. In corresponding plot 1824therefore, frequency domain curve 1846 has ten peaks.

FIG. 18K shows overlay plots 1825 and 1826, which overlay curves1827-1836 and 1837-1846, respectively. As indicated in plot 1825, thevarious time domain curves have different durations. However, in someembodiments, the non-zero durations of the time domain curves are ofsimilar lengths. For an MLO system, the number of signals usedrepresents the number of overlays and the improvement in spectralefficiency. It should be understood that, while ten signals aredisclosed in FIGS. 18A-18K, a greater or lesser quantity of signals maybe used, and that further, a different set of signals, rather than theΨ_(n) signals plotted, may be used.

MLO signals used in a modulation layer have minimum time-bandwidthproducts, which enable improvements in spectral efficiency, and arequadratically integrable. This is accomplished by overlaying multipledemultiplexed parallel data streams, transmitting them simultaneouslywithin the same bandwidth. The key to successful separation of theoverlaid data streams at the receiver is that the signals used withineach symbols period are mutually orthogonal. MLO overlays orthogonalsignals within a single symbol period. This orthogonality prevents ISIand inter-carrier interference (ICI).

Because MLO works in the baseband layer of signal processing, and someembodiments use QAM architecture, conventional wireless techniques foroptimizing air interface, or wireless segments, to other layers of theprotocol stack will also work with MLO. Techniques such as channeldiversity, equalization, error correction coding, spread spectrum,interleaving and space-time encoding are applicable to MLO. For example,time diversity using a multipath-mitigating rake receiver can also beused with MLO. MLO provides an alternative for higher order QAM, whenchannel conditions are only suitable for low order QAM, such as infading channels. MLO can also be used with CDMA to extend the number oforthogonal channels by overcoming the Walsh code limitation of CDMA. MLOcan also be applied to each tone in an OFDM signal to increase thespectral efficiency of the OFDM systems.

Embodiments of MLO systems amplitude modulate a symbol envelope tocreate sub-envelopes, rather than sub-carriers. For data encoding, eachsub-envelope is independently modulated according to N-QAM, resulting ineach sub-envelope independently carrying information, unlike OFDM.Rather than spreading information over many sub-carriers, as is done inOFDM, for MLO, each sub-envelope of the carrier carries separateinformation. This information can be recovered due to the orthogonalityof the sub-envelopes defined with respect to the sum of squares overtheir duration and/or spectrum. Pulse train synchronization or temporalcode synchronization, as needed for CDMA, is not an issue, because MLOis transparent beyond the symbol level. MLO addresses modification ofthe symbol, but since CDMA and TDMA are spreading techniques of multiplesymbol sequences over time. MLO can be used along with CDMA and TDMA.

FIG. 19 illustrates a comparison of MLO signal widths in the time andfrequency domains. Time domain envelope representations 1901-1903 ofsignals SH0-SH3 are illustrated as all having a duration T_(S). SH0-SH3may represent PSI₀-PSI₂, or may be other signals. The correspondingfrequency domain envelope representations are 1905-1907, respectively.SH0 has a bandwidth BW, SH1 has a bandwidth three times BW, and SH2 hasa bandwidth of 5 BW, which is five times as great as that of SH0. Thebandwidth used by an MLO system will be determined, at least in part, bythe widest bandwidth of any of the signals used. The highest ordersignal must set within the available bandwidth. This will set theparameters for each of the lower order signals in each of the layers andenable the signals to fit together without interference. If each layeruses only a single signal type within identical time windows, thespectrum will not be fully utilized, because the lower order signalswill use less of the available bandwidth than is used by the higherorder signals.

FIG. 20A illustrates a spectral alignment of MLO signals that accountsfor the differing bandwidths of the signals, and makes spectral usagemore uniform, using SH0-SH3. Blocks 2001-2004 are frequency domainblocks of an OFDM signal with multiple subcarriers. Block 2003 isexpanded to show further detail. Block 2003 comprises a first layer 2003x comprised of multiple SH0 envelopes 2003 a-2003 o. A second layer 2003y of SH1 envelopes 2003 p-2003 t has one third the number of envelopesas the first layer. In the illustrated example, first layer 2003 x has15 SH0 envelopes, and second layer 2003 y has five SH1 envelopes. Thisis because, since the SH1 bandwidth envelope is three times as wide asthat of SH0, 15 SH0 envelopes occupy the same spectral width as five SH1envelopes. The third layer 2003 z of block 2003 comprises three SH2envelopes 2003 u-2003 w, because the SH2 envelope is five times thewidth of the SH0 envelope.

The total required bandwidth for such an implementation is a multiple ofthe least common multiple of the bandwidths of the MLO signals. In theillustrated example, the least common multiple of the bandwidth requiredfor SH0, SH1, and SH2 is 15 BW, which is a block in the frequencydomain. The OFDM-MLO signal can have multiple blocks, and the spectralefficiency of this illustrated implementation is proportional to(15+5+3)/15.

FIGS. 20B-20C illustrate a situation wherein the frequency domainenvelopes 2020-2024 are each located in a separate layer within a samephysical band width 2025. However, each envelope rather than beingcentered on a same center frequency as shown in FIG. 19 has its owncenter frequency 2026-2030 shifted in order to allow a slided overlay.The purposed of the slided center frequency is to allow better use ofthe available bandwidth and insert more envelopes in a same physicalbandwidth.

Since each of the layers within the MLO signal comprises a differentchannel, different service providers may share a same bandwidth by beingassigned to different MLO layers within a same bandwidth. Thus, within asame bandwidth, service provider one may be assigned to a first MLOlayer, service provider two may be assigned to a second MLO layer and soforth.

FIG. 21 illustrates another spectral alignment of MLO signals, which maybe used alternatively to alignment scheme shown in FIG. 20. In theembodiment illustrated in FIG. 21, the OFDM-MLO implementation stacksthe spectrum of SH0, SH1, and SH2 in such a way that the spectrum ineach layer is utilized uniformly. Layer 2100A comprises envelopes2101A-2101D, which includes both SH0 and SH2 envelopes. Similarly, layer2100C, comprising envelopes 2103A-2103D, includes both SH0 and SH2envelopes. Layer 2100B, however, comprising envelopes 2102A-2102D,includes only SH1 envelopes. Using the ratio of envelope sizes describedabove, it can be easily seen that BW+5 BW=3 BW+3 BW. Thus, for each SH0envelope in layer 2100A, there is one SH2 envelope also in layer 2100Cand two SH1 envelopes in layer 2100B.

Three Scenarios Compared:

1) MLO with 3 Layers defined by:

$\begin{matrix}{{{f_{0}(t)} = {W_{0}e^{\frac{t^{2}}{4}}}},} & {W_{0} = 0.6316} \\{{{f_{1}(t)} = {W_{1}{te}^{\frac{t^{2}}{4}}}},} & {W_{1} \approx 0.6316} \\{{{f_{2}(t)} = {{W_{2}\left( {t^{2} - 1} \right)}e^{\frac{t^{2}}{4}}}},} & {W_{2} \approx 0.4466}\end{matrix}$

(The current FPGA implementation uses the truncation interval of [−6,6].)2) Conventional scheme using rectangular pulse3) Conventional scheme using a square-root raised cosine (SRRC) pulsewith a roll-off factor of 0.5

For MLO pulses and SRRC pulse, the truncation interval is denoted by[−t1, t1] in the following Fig.s. For simplicity, we used the MLO pulsesdefined above, which can be easily scaled in time to get the desiredtime interval (say micro-seconds or nano-seconds). For the SRRC pulse,we fix the truncation interval of [−3T, 3T] where T is the symbolduration for all results presented in this document.

Bandwidth Efficiency

The X-dB bounded power spectral density bandwidth is defined as thesmallest frequency interval outside which the power spectral density(PSD) is X dB below the maximum value of the PSD. The X-dB can beconsidered as the out-of-band attenuation.

The bandwidth efficiency is expressed in Symbols per second per Hertz.The bit per second per Hertz can be obtained by multiplying the symbolsper second per Hertz with the number of bits per symbol (i.e.,multiplying with log 2 M for M-ary QAM).

Truncation of MLO pulses introduces inter-layer interferences (ILI).However, the truncation interval of [−6, 6] yields negligible ILI while[−4, 4] causes slight tolerable ILI. Referring now to FIG. 22, there isillustrated the manner in which a signal, for example a superQAM signal,may be layered to create ILI. FIG. 22 illustrates 3 different superQAMsignals 2202, 2204 and 2206. The superQAM signals 2202-2206 may betruncated and overlapped into multiple layers using QLO in the mannerdescribed herein above. However, as illustrated in FIG. 66, thetruncation of the superQAM signals 2202-2206 that enables the signals tobe layered together within a bandwidth T_(d) 2302 creates a singlesignal 2304 having the interlayer interference between each of thelayers containing a different signal produced by the QLO process. TheILI is caused between a specific bit within a specific layer having aneffect on other bits within another layer of the same symbol.

The bandwidth efficiency of MLO may be enhanced by allowing inter-symbolinterference (ISI). To realize this enhancement, designing transmitterside parameters as well as developing receiver side detection algorithmsand error performance evaluation can be performed. One manner in whichISI may be created is when multilayer signals such as that illustratedin FIG. 23 are overlapped with each other in the manner illustrated inFIG. 24. Multiple signal symbols 2402 are overlapped with each other inorder to enable to enable more symbols to be located within a singlebandwidth. The portions of the signal symbols 2402 that are overlappingcause the creation of ISI. Thus, a specific bit at a specific layer willhave an effect on the bits of nearby symbols.

The QLO transmission and reception system can be designed to have aparticular known overlap between symbols. The system can also bedesigned to calculate the overlaps causing ISI (symbol overlap) and ILI(layer overlay). The ISI and ILI can be expressed in the format of aNM*NM matrix derived from a N*NM matrix. N comprises the number oflayers and M is the number of symbols when considering ISI. Referringnow to FIG. 25, there is illustrated a fixed channel matrix H_(xy) whichis a N*NM matrix. From this we can calculate another matrix which isH_(yx) which is a NM*NM matrix. The ISI and ILI can be canceled by (a)applying a filter of H_(yx) ⁻¹ to the received vector or (b)pre-distorting the transmitted signal by the SVD (singular valuedecomposition) of H_(yx) ⁻¹. Therefore, by determining the matrix H_(xy)of the fixed channel, the signal may be mathematically processed toremove ISL and ILI.

When using orthogonal functions such as Hermite Guassian (HG) functions,the functions are all orthogonal for any permutations of the index ifinfinitely extended. However, when the orthogonal functions aretruncated as discussed herein above, the functions becomepseudo-orthogonal. This is more particularly illustrated in FIG. 26. Inthis case, orthogonal functions are represented along each of the axes.At the intersection of the same orthogonal functions, functions arecompletely correlated and a value of “1” is indicated. Thus, a diagonalof “1” exists with each of the off diagonal elements comprising a “0”since these functions are completely orthogonal with each other. Whentruncated HG choose functions are used the 0 values will not be exactly0 since the functions are no longer orthogonal but arepseudo-orthogonal.

However, the HG functions can be selected in a manner that the functionsare practically orthogonal. This is achieved by selecting the HG signalsin a sequence to achieve better orthogonality. Thus, rather thanselecting the initial three signals in a three signal HG signal sequence(P0 P1 P2), various other sequences that do not necessarily comprise thefirst three signals of the HG sequence may be selected as shown below.

P0 P1 P4 P0 P3 P6 P0 P1 P6 P0 P4 P5 P0 P2 P3 P0 P5 P6 P0 P2 P5 P1 P3 P6P0 P3 P4 P2 P5 P6

Similar selection of sequences may be done to achieve betterorthogonality with two signals, four signals, etc.

The techniques described herein are applicable to a wide variety ofcommunication band environments. They may be applied across the visibleand invisible bands and include RF, Fiber, Freespace optical and anyother communications bands that can benefit from the increased bandwidthprovided by the disclosed techniques.

Application of OAM to Optical Communication

Utilization of OAM for optical communications is based on the fact thatcoaxially propagating light beams with different OAM states can beefficiently separated. This is certainly true for orthogonal modes suchas the LG beam. Interestingly, it is also true for general OAM beamswith cylindrical symmetry by relying only on the azimuthal phase.Considering any two OAM beams with an azimuthal index of l 1 and l 2,respectively:

U ₁(r,θ,z)=A ₁(r,z)exp(il ₁θ)  (12)

where r and z refers to the radial position and propagation distancerespectively, one can quickly conclude that these two beams areorthogonal in the sense that:

$\begin{matrix}{{\int_{0}^{2\; \pi}{U_{1}U_{2}^{*}\ {\theta}}} = \left\{ \begin{matrix}0 & {{{if}\mspace{14mu} _{1}} \neq _{2}} \\{A_{1}A_{2}^{*}} & {{{if}\mspace{14mu} _{1}} = _{2}}\end{matrix} \right.} & (13)\end{matrix}$

There are two different ways to take advantage of the distinctionbetween OAM beams with different l states in communications. In thefirst approach, N different OAM states can be encoded as N differentdata symbols representing “0”, “1”, . . . , “N−1”, respectively. Asequence of OAM states sent by the transmitter therefore represents datainformation. At the receiver, the data can be decoded by checking thereceived OAM state. This approach seems to be more favorable to thequantum communications community, since OAM could provide for theencoding of multiple bits (log 2(N)) per photon due to the infinitelycountable possibilities of the OAM states, and so could potentiallyachieve a higher photon efficiency. The encoding/decoding of OAM statescould also have some potential applications for on-chip interconnectionto increase computing speed or data capacity.

The second approach is to use each OAM beam as a different data carrierin an SDM (Spatial Division Multiplexing) system. For an SDM system, onecould use either a multi-core fiber/free space laser beam array so thatthe data channels in each core/laser beam are spatially separated, oruse a group of orthogonal mode sets to carry different data channels ina multi-mode fiber (MMF) or in free space. Greater than 1 petabit/s datatransmission in a multi-core fiber and up to 6 linearly polarized (LP)modes each with two polarizations in a single core multi-mode fiber hasbeen reported. Similar to the SDM using orthogonal modes, OAM beams withdifferent states can be spatially multiplexed and demultiplexed, therebyproviding independent data carriers in addition to wavelength andpolarization. Ideally, the orthogonality of OAM beams can be maintainedin transmission, which allows all the data channels to be separated andrecovered at the receiver. A typical embodiments of OAM multiplexing isconceptually depicted in FIG. 27. An obvious benefit of OAM multiplexingis the improvement in system spectral efficiency, since the samebandwidth can be reused for additional data channels.

OAM Beam Generation and Detection

Many approaches for creating OAM beams have been proposed anddemonstrated. One could obtain a single or multiple OAM beams directlyfrom the output of a laser cavity, or by converting a fundamentalGaussian beam into an OAM beam outside a cavity. The converter could bea spiral phase plate, diffractive phase holograms, metalmaterials,cylindrical lens pairs, q-plates or fiber structures. There are alsodifferent ways to detect an OAM beam, such as using a converter thatcreates a conjugate helical phase, or using a plasmonic detector.

Mode Conversion Approaches

Referring now to FIG. 28, among all external-cavity methods, perhaps themost straightforward one is to pass a Gaussian beam through a coaxiallyplaced spiral phase plate (SPP) 2802. An SPP 2802 is an optical elementwith a helical surface, as shown in FIG. 12E. To produce an OAM beamwith a state of l, the thickness profile of the plate should be machinedas lλθ/2π(n−1), where n is the refractive index of the medium. Alimitation of using an SPP 2802 is that each OAM state requires adifferent specific plate. As an alternative, reconfigurable diffractiveoptical elements, e.g., a pixelated spatial light modulator (SLM) 2804,or a digital micro-mirror device can be programmed to function as anyrefractive element of choice at a given wavelength. As mentioned above,a helical phase profile exp(ilθ) converts a linearly polarized Gaussianlaser beam into an OAM mode, whose wave front resembles an l-foldcorkscrew 2806, as shown at 2804. Importantly, the generated OAM beamcan be easily changed by simply updating the hologram loaded on the SLM2804. To spatially separate the phase-modulated beam from thezeroth-order non-phase-modulated reflection from the SLM, a linear phaseramp is added to helical phase code (i.e., a “fork”-like phase pattern2808 to produce a spatially distinct first-order diffracted OAM beam,carrying the desired charge. It should also be noted that theaforementioned methods produce OAM beams with only an azimuthal indexcontrol. To generate a pure LG_(l,p) mode, one must jointly control boththe phase and the intensity of the wavefront. This could be achievedusing a phase-only SLM with a more complex phase hologram.

Some novel material structures, such as metal-surface, can also be usedfor OAM generation. A compact metal-surface could be made into a phaseplate by manipulation of the structure caused spatial phase response. Asshown in FIGS. 29A and 29B, a V-shaped antenna array 2902 is fabricatedon the metal surface 2904, each of which is composed of two arms 2906,2908 connected at one end 2910. A light reflected by this plate wouldexperience a phase change ranging from 0 to 2π, determined by the lengthof the arms and angle between two arms. To generate an OAM beam, thesurface is divided into 8 sectors 2912, each of which introduces a phaseshift from 0 to 77π/4 with a step of π/4. The OAM beam with l=+1 isobtained after the reflection, as shown in FIG. 29C.

Referring now to FIG. 30, another interesting liquid crystal-baseddevice named “q-plate” 3002 is also used as a mode converter whichconverts a circularly polarized beam 3004 into an OAM beam 3006. Aq-plate is essentially a liquid crystal slab with a uniform birefringentphase retardation of π and a spatially varying transverse optical axis3008 pattern. Along the path circling once around the center of theplate, the optical axis of the distributed crystal elements may have anumber of rotations defined by the value of q. A circularly polarizedbeam 3004 passing through this plate 3002 would experience a helicalphase change of exp (ilθ) with l=2q, as shown in FIG. 30.

Note that almost all the mode conversion approaches can also be used todetect an OAM beam. For example, an OAM beam can be converted back to aGaussian-like non-OAM beam if the helical phase front is removed, e.g.,by passing the OAM beam through a conjugate SPP or phase hologram.

Intra-Cavity Approaches

Referring now to FIG. 31, OAM beams are essentially higher order modesand can be directly generated from a laser resonator cavity. Theresonator 3100 supporting higher order modes usually produce the mixtureof multiple modes 3104, including the fundamental mode. In order toavoid the resonance of fundamental Gaussian mode, a typical approach isto place an intra-cavity element 3106 (spiral phase plate, tiled mirror)to force the oscillator to resonate on a specific OAM mode. Otherreported demonstrations include the use of an annular shaped beam aslaser pump, the use of thermal lensing, or by using a defect spot on oneof the resonator mirrors.

OAM Beams Multiplexing and Demultiplexing

One of the benefits of OAM is that multiple coaxially propagating OAMbeams with different l states provide additional data carriers as theycan be separated based only on the twisting wavefront. Hence, one of thecritical techniques is the efficient multiplexing/demultiplexing of OAMbeams of different l states, where each carries an independent datachannel and all beams can be transmitted and received using a singleaperture pair. Several multiplexing and demultiplexing techniques havebeen demonstrated, including the use of an inverse helical phasehologram to down-convert the OAM into a Gaussian beam, a mode sorter,free-space interferometers, a photonic integrated circuit, and q-plates.Some of these techniques are briefly described below.

Beam Splitter and Inverse Phase Hologram

A straightforward way of multiplexing is simply to use cascaded 3-dBbeam splitters (BS) 3202. Each BS 3202 can coaxially multiplex two beams3203 that are properly aligned, and cascaded N BSs can multiplex N+1independent OAM beams at most, as shown in FIG. 32. Similarly, at thereceiver end, the multiplexed beam 3205 is divided into four copies 3204by BS 3202. To demultiplex the data channel on one of the beams (e.g.,with 1=l_i), a phase hologram 3206 with a spiral charge of

−1

_i is applied to all the multiplexed beams 3204. As a result, thehelical phase on the target beam is removed, and this beam evolves intoa fundamental Gaussian beam, as shown in FIG. 33. The down-convertedbeam can be isolated from the other beams, which still have helicalphase fronts by using a spatial mode filter 3308 (e.g., a single modefiber only couples the power of the fundamental Gaussian mode due to themode matching theory). Accordingly, each of the multiplexed beams 3304can be demultiplexed by changing the spiral phase hologram 3306.Although this method is very power-inefficient since the BSs 3302 andthe spatial mode filter 3306 cause a lot of power loss, it was used inthe initial lab demonstrations of OAM multiplexing/demultiplexing, dueto the simplicity of understanding and the reconfigurability provided byprogrammable SLMs.

Optical Geometrical Transformation-Based Mode Sorter

Referring now to FIG. 34, another method of multiplexing anddemultiplexing, which could be more power-efficient than the previousone (using beam splitters), is the use of an OAM mode sorter. This modesorter usually comprises three optical elements, including a transformer3402, a corrector 3404, and a lens 3406, as shown in FIG. 34. Thetransformer 3402 performs a geometrical transformation of the input beamfrom log-polar coordinates to Cartesian coordinates, such that theposition (x,y) in the input plane is mapped to a new position (u,v) inthe output plane, where

${{u = {{- a}\; {\ln \left( \frac{\sqrt{x^{2} + y^{2}}}{b} \right)}}},\mspace{11mu} \; {and}}\mspace{14mu}$v = a  arctan (y/x).

Here, a and b are scaling constants. The corrector 3404 compensates forphase errors and ensures that the transformed beam is collimated.Considering an input OAM beam with a ring-shaped beam profile, it can beunfolded and mapped into a rectangular-shaped plane wave with a tiltedphase front. Similarly, multiple OAM beams having different 1 stateswill be transformed into a series of plane waves each with a differentphase tilt. A lens 3406 focuses these tilted plane waves into spatiallyseparated spots in the focal plane such that all the OAM beams aresimultaneously demultiplexed. As the transformation is reciprocal, ifthe mode sorter is used in reverse it can become a multiplexer for OAM.A Gaussian beam array placed in the focal plane of the lens 3406 isconverted into superimposed plane waves with different tilts. Thesebeams then pass through the corrector and the transformer sequentiallyto produce properly multiplexed OAM beams.

Free Space Communications

The first proof-of-concept experiment using OAM for free spacecommunications transmitted eight different OAM states each representinga data symbol one at a time. The azimuthal index of the transmitted OAMbeam is measured at the receiver using a phase hologram modulated with abinary grating. To effectively use this approach, fast switching isrequired between different OAM states to achieve a high data rate.Alternatively, classic communications using OAM states as data carrierscan be multiplexed at the transmitter, co-propagated through a freespace link, and demultiplexed at a receiver. The total data rate of afree space communication link has reached 100 Tbit/s or even beyond byusing OAM multiplexing. The propagation of OAM beams through a realenvironment (e.g., across a city) is also under investigation.

Basic Link Demonstrations

Referring now to FIGS. 35-36B, initial demonstrates of using OAMmultiplexing for optical communications include free space links using aGaussian beam and an OAM beam encoded with OOK data. Four monochromaticGaussian beams each carrying an independent 50.8 Gbit/s (4×12.7 Gbit/s)16-QAM signal were prepared from an IQ modulator and free-spacecollimators. The beams were converted to OAM beams with l=−8, +10, +12and −14, respectively, using 4 SLMs each loaded with a helical phasehologram, as shown in FIG. 30A. After being coaxially multiplexed usingcascaded 3 dB-beam splitters, the beams were propagated through ˜1 mdistance in free-space under lab conditions. The OAM beams were detectedone at a time, using an inverse helical phase hologram and a fibercollimator together with a SMF. The 16-QAM data on each channel wassuccessfully recovered, and a spectral efficiency of 12.8 bit/s/Hz inthis data link was achieved, as shown in FIGS. 36A and 36B.

A following experiment doubled the spectral efficiency by adding thepolarization multiplexing into the OAM-multiplexed free-space data link.Four different OAM beams (l=+4, +8, −8, +16) on each of two orthogonalpolarizations (eight channels in total) were used to achieve a Terabit/stransmission link. The eight OAM beams were multiplexed anddemultiplexed using the same approach as mentioned above. The measuredcrosstalk among channels carried by the eight OAM beams is shown inTable 1, with the largest crosstalk being ˜−18.5 dB. Each of the beamswas encoded with a 42.8 Gbaud 16-QAM signal, allowing a total capacityof ˜1.4 (42.8×4×4×2) Tbit/s.

TABLE 1 OAM₊₄ OAM₊₈ OAM⁻⁸ OAM₊₁₆ Measured Crosstalk X-Pol. Y-Pol. X-Pol.Y-Pol. X-Pol. Y-Pol. X-Pol. Y-Pol. OAM₊₄ (dB) X-Pol. −23.2 −26.7 −30.8−30.5 −27.7 −24.6 −30.1 Y-Pol. −25.7 OAM₊₈ (dB) X-Pol. −26.6 −23.5 −21.6−18.9 −25.4 −23.9 −28.8 Y-Pol. −25.0 OAM⁻⁸ (dB) X-Pol. −27.5 −33.9 −27.6−30.8 −20.5 −26.5 −21.6 Y-Pol. −26.8 OAM₊₁₆ (dB) X-Pol. −24.5 −31.2−23.7 −23.3 −25.8 −26.1 −30.2 Y-Pol. −24.0 Total from other OAMs * (dB)−21.8 −21.0 −21.2 −21.4 −18.5 −21.2 −22.2 −20.7

The capacity of the free-space data link was further increased to 100Tbit/s by combining OAM multiplexing with PDM (phase divisionmultiplexing) and WDM (wave division multiplexing). In this experiment,24 OAM beams (l=±4, ±7, ±10, ±13, ±16, and ±19, each with twopolarizations) were prepared using 2 SLMs, the procedures for which areshown in FIG. 37A at 3702-3706. Specifically, one SLM generated asuperposition of OAM beams with l=+4, +10, and +16, while the other SLMgenerated another set of three OAM beams with l=+7, +13, and +19 (FIG.37A). These two outputs were multiplexed together using a beam splitter,thereby multiplexing six OAM beams: l=+4, +7, +10, +13, +16, and +19(FIG. 37A). Secondly, the six multiplexed OAM beams were split into twocopies. One copy was reflected five times by three mirrors and two beamsplitters, to create another six OAM beams with inverse charges (FIG.37B). There was a differential delay between the two light paths tode-correlate the data. These two copies were then combined again toachieve 12 multiplexed OAM beams with l=±4, ±7, ±10, ±13, ±16, and ±19(FIG. 37B). These 12 OAM beams were split again via a beam splitter. Oneof these was polarization-rotated by 90 degrees, delayed by ˜33 symbols,and then recombined with the other copy using a polarization beamsplitter (PBS), finally multiplexing 24 OAM beams (with f=±4, ±7, ±10,±13, ±16, and ±19 on two polarizations). Each of the beam carried a WDMsignal comprising 100 GHz-spaced 42 wavelengths (1,536.34-1,568.5 nm),each of which was modulated with 100 Gbit/s QPSK data. The observedoptical spectrum of the WDM signal carried on one of the demultiplexedOAM beams (l=+10).

Atmospheric Turbulence Effects on OAM Beams

One of the critical challenges for a practical free-space opticalcommunication system using OAM multiplexing is atmospheric turbulence.It is known that inhomogeneities in the temperature and pressure of theatmosphere lead to random variations in the refractive index along thetransmission path, and can easily distort the phase front of a lightbeam. This could be particularly important for OAM communications, sincethe separation of multiplexed OAM beams relies on the helicalphase-front. As predicted by simulations in the literature, theserefractive index inhomogeneities may cause inter-modal crosstalk amongdata channels with different OAM states.

The effect of atmospheric turbulence is also experimentally evaluated.For the convenience of estimating the turbulence strength, one approachis to emulate the turbulence in the lab using an SLM or a rotating phaseplate. FIG. 38A illustrates an emulator built using a thin phase screenplate 3802 that is mounted on a rotation stage 3804 and placed in themiddle of the optical path. The pseudo-random phase distributionmachined on the plate 3802 obeys Kolmogorov spectrum statistics, whichare usually characterized by a specific effective Fried coherence lengthr0. The strength of the simulated turbulence 3806 can be varied eitherby changing to a plate 3802 with a different r0, or by adjusting thesize of the beam that is incident on the plate. The resultant turbulenceeffect is mainly evaluated by measuring the power of the distorted beamdistributed to each OAM mode using an OAM mode sorter. It was foundthat, as the turbulence strength increases, the power of the transmittedOAM mode would leak to neighboring modes and tend to be equallydistributed among modes for stronger turbulence. As an example, FIG. 38Bshows the measured average power (normalized) 1=3 beam under differentemulated turbulence conditions. It can be seen that the majority of thepower is still in the transmitted OAM mode 3808 under weak turbulence,but it spreads to neighboring modes as the turbulence strengthincreases.

Turbulence Effects Mitigation Techniques

One approach to mitigate the effects of atmospheric turbulence on OAMbeams is to use an adaptive optical (AO) system. The general idea of anAO system is to measure the phase front of the distorted beam first,based on which an error correction pattern can be produced and can beapplied onto the beam transmitter to undo the distortion. As for OAMbeams with helical phase fronts, it is challenging to directly measurethe phase front using typical wavefront sensors due to the phasesingularity. A modified AO system can overcome this problem by sending aGaussian beam as a probe beam to sense the distortion, as shown in FIG.39A. Due to the fact that turbulence is almost independent of the lightpolarization, the probe beam is orthogonally polarized as compared toall other beams for the sake of convenient separation at beam separator3902. The correction phase pattern can be derived based on the probebeam distortion that is directly measured by a wavefront sensor 3804. Itis noted that this phase correction pattern can be used tosimultaneously compensate multiple coaxially propagating OAM beams. FIG.39 at 3910-3980 illustrate the intensity profiles of OAM beams with 1=1,5 and 9, respectively, for a random turbulence realization with andwithout mitigation. From the far-field images, one can see that thedistorted OAM beams (upper), up to 1=9, were partially corrected, andthe measured power distribution also indicates that the channelcrosstalk can be reduced.

Another approach for combating turbulence effects is to partially movethe complexity of optical setup into the electrical domain, and usedigital signal processing (DSP) to mitigate the channel crosstalk. Atypical DSP method is the multiple-input-multiple-output (MIMO)equalization, which is able to blindly estimate the channel crosstalkand cancel the interference. The implementation of a 4×4 adaptive MIMOequalizer in a four-channel OAM multiplexed free space optical linkusing heterodyne detection may be used. Four OAM beams (l=+2, +4, +6 and+8), each carrying 20 Gbit/s QPSK data, were collinearly multiplexed andpropagated through a weak turbulence emulated by the rotating phaseplate under laboratory condition to introduce distortions. Afterdemultiplexing, four channels were coherently detected and recordedsimultaneously. The standard constant modulus algorithm is employed inaddition to the standard procedures of coherent detection to equalizethe channel interference. Results indicate that MIMO equalization couldbe helpful to mitigate the crosstalk caused by either turbulence orimperfect mode generation/detection, and improve both error vectormagnitude (EVM) and the bit-error-rate (BER) of the signal in anOAM-multiplexed communication link. MIMO DSP may not be universallyuseful as outage could happen in some scenarios involving free spacedata links. For example, the majority power of the transmitted OAM beamsmay be transferred to other OAM states under a strong turbulence withoutbeing detected, in which case MIMO would not help to improve the systemperformance.

OAM Free Space Link Design Considerations

To date, most of the experimental demonstrations of opticalcommunication links using OAM beams took place in the lab conditions.There is a possibility that OAM beams may also be used in a free spaceoptical communication link with longer distances. To design such a datalink using OAM multiplexing, several important issues such as beamdivergence, aperture size and misalignment of two transmitter andreceiver, need to be resolved. To study how those parameters affect theperformance of an OAM multiplexed system, a simulation model wasdescribed by Xie et al, the schematic setup of which is shown in FIG.40. Each of the different collimated Gaussian beams 4002 at the samewavelength is followed by a spiral phase plate 4004 with a unique orderto convert the Gaussian beam into a data-carrying OAM beam. Differentorders of OAM beams are then multiplexed at multiplexor 4006 to form aconcentric-ring-shape and coaxially propagate from transmitter 4008through free space to the receiver aperture located at a certainpropagation distance. Propagation of multiplexed OAM beams isnumerically propagated using the Kirchhoff-Fresnel diffraction integral.To investigate the signal power and crosstalk effect on neighboring OAMchannels, power distribution among different OAM modes is analyzedthrough a modal decomposition approach, which corresponds to the casewhere the received OAM beams are demultiplexed without power loss andthe power of a desired OAM channel is completely collected by itsreceiver 4010.

Beam Divergence

For a communication link, it is generally preferable to collect as muchsignal power as possible at the receiver to ensure a reasonablesignal-to-noise ratio (SNR). Based on the diffraction theory, it isknown that a collimated OAM beam diverges while propagating in freespace. Given the same spot size of three cm at the transmitter, an OAMbeam with a higher azimuthal index diverges even faster, as shown inFIG. 41A. On the other hand, the receiving optical element usually has alimited aperture size and may not be able to collect all of the beampower. The calculated link power loss as a function of receiver aperturesize is shown in FIG. 41B, with different transmission distances andvarious transmitted beam sizes. Unsurprisingly, the power loss of a 1-kmlink is higher than that of a 100-m link under the same transmitted beamsize due to larger beam divergence. It is interesting to note that asystem with a transmitted beam size of 3 cm suffers less power loss thanthat of 1 cm and 10 cm over a 100-m link. The 1-cm transmitted beamdiverges faster than the 3 cm beam due to its larger diffraction.However, when the transmitted beam size is 10 cm, the geometricalcharacteristics of the beam dominate over the diffraction, thus leadinglarger spot size at the receiver than the 3 cm transmitted beam. Atrade-off between the diffraction, geometrical characteristics and thenumber of OAMs of the beam therefore needs to be carefully considered inorder to achieve a proper-size received beam when designing a link.

Misalignment Tolerance

Referring now to FIGS. 42A-42C, besides the power loss due tolimited-size aperture and beam divergence, another issue that needsfurther discussion is the potential misalignment between the transmitterand the receiver. In an ideal OAM multiplexed communication link,transmitter and receiver would be perfectly aligned, (i.e., the centerof the receiver would overlap with the center of the transmitted beam4202, and the receiver plane 4204 would be perpendicular to the lineconnecting their centers, as shown in FIG. 42A). However, due todifficulties in aligning because of substrate distances, and jitter andvibration of the transmitter/receiver platform, the transmitter andreceiver may have relative lateral shift (i.e., lateral displacement)(FIG. 42B) or angular shift (i.e., receiver angular error) (FIG. 42C).Both types of misalignment may lead to degradation of systemperformance.

Focusing on a link distance of 100 m, FIGS. 43A and 43B show the powerdistribution among different OAM modes due to lateral displacement andreceiver angular error when only l=+3 is transmitted with a transmittedbeam size of 3 cm. In order to investigate the effect of misalignment,the receiver aperture size is chosen to be 10 cm, which could cover thewhole OAM beam at the receiver. As the lateral displacement or receiverangular error increases, power leakage to other modes (i.e., channelcrosstalk) increases while the power on l=+3 state decreases. This isbecause larger lateral displacement or receiver angular causes largerphase profile mismatch between the received OAM beams and receiver. Thepower leakage to l=+1 and l=+5 is greater than that of l=+2 and l=+3 dueto their larger mode spacing with respect to l=+3. Therefore, a systemwith larger mode spacing (which also uses higher order OAM statessuffers less crosstalk. However, such a system may also have a largerpower loss due to the fast divergence of higher order OAM beams, asdiscussed above. Clearly, this trade-off between channel crosstalk andpower loss shall be considered when choosing the mode spacing in aspecific OAM multiplexed link.

Referring now to FIG. 44, there is a bandwidth efficiency comparisonversus out of band attenuation (X-dB) where quantum level overlay pulsetruncation interval is [−6,6] and the symbol rate is ⅙. Referring alsoto FIG. 45, there is illustrated the bandwidth efficiency comparisonversus out of band attenuation (X-dB) where quantum level overlay pulsetruncation interval is [−6,6] and the symbol rate is ¼.

The QLO signals are generated from the Physicist's special Hermitefunctions:

${{f_{n}\left( {t,\alpha} \right)} = {\sqrt{\frac{\alpha}{\sqrt{\pi \;}{n!}2^{n}}}{H_{n}\left( {\alpha \; t} \right)}e^{- \frac{\alpha^{2}t^{2}}{2}}}},{\alpha > 0}$

Note that the initial hardware implementation is using

$\alpha = \frac{1}{\sqrt{2}}$

and for consistency with his part,

$\alpha = \frac{1}{\sqrt{2}}$

is used in all Fig.s related to the spectral efficiency.

Let the low-pass-equivalent power spectral density (PSD) of the combinedQLO signals be X(f) and its bandwidth be B. Here the bandwidth isdefined by one of the following criteria.

ACLR1 (First Adjacent Channel Leakage Ratio) in dBc equals:

${{ACLR}\; 1} = \frac{\int_{B/2}^{3{B/2}}{{X(f)}{df}}}{\int_{- \infty}^{\infty}{{X(f)}{df}}}$

ACLR2 (Second Adjacent Channel Leakage Ratio) in dBc equals:

${{ACLR}\; 2} = \frac{\int_{3{B/2}}^{5{B/2}}{{X(f)}{df}}}{\int_{- \infty}^{\infty}{{X(f)}{df}}}$

Out-of-Band Power to Total Power Ratio is:

$\frac{2{\int_{B/2}^{\infty}{{X(f)}{df}}}}{\int_{- \infty}^{\infty}{{X(f)}{df}}}$

The Band-Edge PSD in dBc/100 kHz equals:

$\frac{\int_{B/2}^{\frac{B}{2} + 10^{5}}{{X(f)}{df}}}{\int_{- \infty}^{\infty}{{X(f)}{df}}}$

Referring now to FIG. 46 there is illustrated a performance comparisonusing ACLR1 and ACLR2 for both a square root raised cosine scheme and amultiple layer overlay scheme. Line 4602 illustrates the performance ofa square root raised cosine 4602 using ACLR1 versus an MLO 4604 usingACLR1. Additionally, a comparison between a square root raised cosine4606 using ACLR2 versus MLO 4608 using ACLR2 is illustrated. Table 2illustrates the performance comparison using ACLR.

TABLE 2 Criteria: ACLR1 ≦−30 dBc per bandwidth Spectral Efficiency ACLR2≦−43 dBc per bandwidth (Symbol/sec/Hz) Gain SRRC [−8T, 8T] β = 0.220.8765 10 Symbol Duration N Layers (Tmol) QLO N = 3 Tmol = 4 1.1331.2926 [−8, 8] N = 4 Tmol = 5 1.094 1.2483 Tmol = 4 1.367 1.5596 N = 10Tmol = 8 1.185 1.3520 Tmol = 7 1.355 1.5459 Tmol = 6 1.580 1.8026 Tmol =5 1.896 2.1631 Tmol = 4 2.371 2.7053

Referring now to FIG. 47, there is illustrated a performance comparisonbetween a square root raised cosine 4702 and a MLO 4704 usingout-of-band power. Referring now also to Table 3, there is illustrated amore detailed comparison of the performance using out-of-band power.

TABLE 3 Table 3: Performance Comparison Using Out-of-Band PowerCriterion: Spectral Efficiency Out-of-band Power/Total Power ≦−30 dB(Symbol/sec/Hz) Gain SRRC [−8T, 8T] β = 0.22 0.861 1.0 Symbol Duration NLayers (Tmol) QLO N = 3 Tmol = 4 1.080 1.2544 [−8, 8] N = 4 Tmol = 51.049 1.2184 Tmol = 4 1.311 1.5226 N = 10 Tmol = 8 1.152 1.3380 Tmol = 71.317 1.5296 Tmol = 6 1.536 1.7840 Tmol = 5 1.844 2.1417 Tmol = 4 2.3052.6771

Referring now to FIG. 48, there is further provided a performancecomparison between a square root raised cosine 4802 and a MLO 4804 usingband-edge PSD. A more detailed illustration of the performancecomparison is provided in Table 4.

TABLE 4 Table 4 Performance Comparison Using Band-Edge PSD Criterion:Spectral Efficiency Band-Edge PSD = −50 dBc/100 kHz (Symbol/sec/Hz) GainSRRC [−8T, 8T] β = 0.22 0.810 1.0 Symbol Duration N Layers (Tmol) QLO N= 3 Tmol = 4 0.925 1.1420 [−8, 8] N = 4 Tmol = 5 0.912 1.1259 Tmol = 41.14 1.4074 N = 10 Tmol = 8 1.049 1.2951 Tmol = 7 1.198 1.4790 Tmol = 61.398 1.7259 Tmol = 5 1.678 2.0716 Tmol = 4 2.697 2.5889

Referring now to FIGS. 49 and 50, there are more particularlyillustrated the transmit subsystem (FIG. 49) and the receiver subsystem(FIG. 50). The transceiver is realized using basic building blocksavailable as Commercially Off The Shelf products. Modulation,demodulation and Special Hermite correlation and de-correlation areimplemented on a FPGA board. The FPGA board 5002 at the receiver 5000estimated the frequency error and recovers the data clock (as well asdata), which is used to read data from the analog-to-digital (ADC) board5006. The FGBA board 5000 also segments the digital I and Q channels.

On the transmitter side 4900, the FPGA board 4902 realizes the specialhermite correlated QAM signal as well as the necessary control signalsto control the digital-to-analog (DAC) boards 4904 to produce analog I&Qbaseband channels for the subsequent up conversion within the directconversion quad modulator 4906. The direct conversion quad modulator4906 receives an oscillator signal from oscillator 4908.

The ADC 5006 receives the I&Q signals from the quad demodulator 5008that receives an oscillator signal from 5010.

Neither power amplifier in the transmitter nor an LNA in the receiver isused since the communication will take place over a short distance. Thefrequency band of 2.4-2.5 GHz (ISM band) is selected, but any frequencyband of interest may be utilized.

MIMO uses diversity to achieve some incremental spectral efficiency.Each of the signals from the antennas acts as an independent orthogonalchannel. With QLO, the gain in spectral efficiency comes from within thesymbol and each QLO signal acts as independent channels as they are allorthogonal to one another in any permutation. However, since QLO isimplemented at the bottom of the protocol stack (physical layer), anytechnologies at higher levels of the protocol (i.e. Transport) will workwith QLO. Therefore one can use all the conventional techniques withQLO. This includes RAKE receivers and equalizers to combat fading,cyclical prefix insertion to combat time dispersion and all othertechniques using beam forming and MIMO to increase spectral efficiencyeven further.

When considering spectral efficiency of a practical wirelesscommunication system, due to possibly different practical bandwidthdefinitions (and also not strictly bandlimited nature of actual transmitsignal), the following approach would be more appropriate.

Referring now to FIG. 51, consider the equivalent discrete time system,and obtain the Shannon capacity for that system (will be denoted by Cd).Regarding the discrete time system, for example, for conventional QAMsystems in AWGN, the system will be:

y[n]=a x[n]+w[n]

where a is a scalar representing channel gain and amplitude scaling,x[n] is the input signal (QAM symbol) with unit average energy (scalingis embedded in a), y[n] is the demodulator (matched filter) outputsymbol, and index n is the discrete time index.

The corresponding Shannon capacity is:

C _(d)=log₂(1+|a| ²/σ²)

where σ2 is the noise variance (in complex dimension) and |a|2/σ2 is theSNR of the discrete time system.

Second, compute the bandwidth W based on the adopted bandwidthdefinition (e.g., bandwidth defined by −40 dBc out of band power). Ifthe symbol duration corresponding to a sample in discrete time (or thetime required to transmit C_(d) bits) is T, then the spectral efficiencycan be obtained as:

C/W=C _(d)/(T W)bps/Hz

In discrete time system in AWGN channels, using Turbo or similar codeswill give performance quite close to Shannon limit C_(d). Thisperformance in discrete time domain will be the same regardless of thepulse shape used. For example, using either SRRC (square root raisedcosine) pulse or a rectangle pulse gives the same C_(d) (or C_(d)/T).However, when we consider continuous time practical systems, thebandwidths of SRRC and the rectangle pulse will be different. For atypical practical bandwidth definition, the bandwidth for a SRRC pulsewill be smaller than that for the rectangle pulse and hence SRRC willgive better spectral efficiency. In other words, in discrete time systemin AWGN channels, there is little room for improvement. However, incontinuous time practical systems, there can be significant room forimprovement in spectral efficiency.

Referring now to FIG. 52, there is illustrated a PSD plot (BLANK) ofMLO, modified MLO (MMLO) and square root raised cosine (SRRC). From theillustration in FIG. 30, demonstrates the better localization propertyof MLO. An advantage of MLO is the bandwidth. FIG. 30 also illustratesthe interferences to adjacent channels will be much smaller for MLO.This will provide additional advantages in managing, allocating orpackaging spectral resources of several channels and systems, andfurther improvement in overall spectral efficiency. If the bandwidth isdefined by the −40 dBc out of band power, the within-bandwidth PSDs ofMLO and SRRC are illustrated in FIG. 53. The ratio of the bandwidths isabout 1.536. Thus, there is significant room for improvement in spectralefficiency.

Modified MLO systems are based on block-processing wherein each blockcontains N MLO symbols and each MLO symbol has L layers. MMLO can beconverted into parallel (virtual) orthogonal channels with differentchannel SNRs as illustrated in FIG. 54. The outputs provide equivalentdiscrete time parallel orthogonal channels of MMLO.

Referring now to FIG. 55, there are illustrated four MLO symbols thatare included in a single block 5500. The four symbols 5502-5508 arecombined together into the single block 5500. The adjacent symbols5502-5508 each have an overlapping region 5510. This overlapping region5510 causes intersymbol interference between the symbols which must beaccounted for when processing data streams.

Note that the intersymbol interference caused by pulse overlapping ofMLO has been addressed by the parallel orthogonal channel conversion. Asan example, the power gain of a parallel orthogonal virtual channel ofMMLO with three layers and 40 symbols per block is illustrated in FIG.56. FIG. 56 illustrates the channel power gain of the parallelorthogonal channels of MMLO with three layers and T_(sim)=3. By applyinga water filling solution, an optimal power distribution across theorthogonal channels for a fixed transmit power may be obtained. Thetransmit power on the k^(th) orthogonal channel is denoted by P_(k).Then the discrete time capacity of the MMLO can be given by:

$C_{d} = {\sum\limits_{k = 1}^{k}\; {{\log_{2}\left( {1 + \frac{P_{k}{a_{k}}^{2}}{\sigma_{k}^{2}}} \right)}\mspace{14mu} {bits}\mspace{14mu} {per}\mspace{14mu} {block}}}$

Note that K depends on the number of MLO layers, the number of MLOsymbols per block, and MLO symbol duration.

For MLO pulse duration defined by [−t₁, t₁], and symbol durationT_(mlo), the MMLO block length is:

T _(block)=(N−1)T _(mlo)+2t ₁

Suppose the bandwidth of MMLO signal based on the adopted bandwidthdefinition (ACLR, OBP, or other) is W_(mmlo), then the practicalspectral efficiency of MMLO is given by:

$\frac{C_{d}}{W_{mmlo}T_{block}} = {\frac{1}{W_{mmlo}\left\{ {{\left( {N - 1} \right)T_{mlo}} + {2t_{1}}} \right\}}{\sum\limits_{k = 1}^{K}\; {{{l{og}}_{2}\left( {1 + \frac{P_{k}{a_{k}}^{2}}{\sigma_{k}^{2}}} \right)}\frac{bps}{Hz}}}}$

FIGS. 57-58 show the spectral efficiency comparison of MMLO with N=40symbols per block, L=3 layers, T_(mlo)=3, t₁=8, and SRRC with duration[−8T, 8T], T=1, and the roll-off factor β=0.22, at SNR of 5 dB. Twobandwidth definitions based on ACLR1 (first adjacent channel leakagepower ratio) and OBP (out of band power) are used.

FIGS. 59-60 show the spectral efficiency comparison of MMLO with L=4layers. The spectral efficiencies and the gains of MMLO for specificbandwidth definitions are shown in the following tables.

TABLE 5 Spectral Efficiency (bps/Hz) Gain with based on ACLR1 reference≦30 dBc per bandwidth to SRRC SRRC 1.7859 1 MMLO (3 layers, Tmlo = 3)2.7928 1.5638 MMLO (4 layers, Tmlo = 3) 3.0849 1.7274

TABLE 6 Spectral Efficiency (bps/Hz) based on Gain with reference OBP≦−40 dBc to SRRC SRRC 1.7046 1 MMLO (3 layers, Tmlo = 3) 2.3030 1.3510MMLO (4 layers, Tmlo = 3) 2.6697 1.5662

Referring now to FIGS. 61 and 62, there are provided basic blockdiagrams of low-pass-equivalent MMLO transmitters (FIG. 61) andreceivers (FIG. 62). The low-pass-equivalent MMLO transmitter 6100receives a number of input signals 6102 at a block-based transmitterprocessing 6104. The transmitter processing outputs signals to theSH(L−1) blocks 6106 which produce the I&Q outputs. These signals arethen all combined together at a combining circuit 6108 for transmission.

Within the baseband receiver (FIG. 62) 6200, the received signal isseparated and applied to a series of match filters 6202. The outputs ofthe match filters are then provided to the block-based receiverprocessing block 6204 to generate the various output streams.

Consider a block of N MLO-symbols with each MLO symbol carrying Lsymbols from L layers. Then there are NL symbols in a block. Define c(m,n)=symbol transmitted by the m-th MLO layer at the n-th MLO symbol.Write all NL symbols of a block as a column vector as follows:c=[c(0,0), c(1,0), . . . , c(L−1, 0), c(0,1), c(1,1), . . . , c(L−1, 1),. . . , c(L−1, N−1)]T. Then the outputs of the receiver matched filtersfor that transmitted block in an AWGN channel, defined by the columnvector y of length NL, can be given as y=H c+n, where H is an NL×NLmatrix representing the equivalent MLO channel, and n is a correlatedGaussian noise vector.

By applying SVD to H, we have H=U D VH where D is a diagonal matrixcontaining singular values. Transmitter side processing using V and thereceiver side processing UH, provides an equivalent system with NLparallel orthogonal channels, (i.e., y=H Vc+n and UH y=Dc+UH n). Theseparallel channel gains are given by diagonal elements of D. The channelSNR of these parallel channels can be computed. Note that by thetransmit and receive block-based processing, we obtain parallelorthogonal channels and hence the ISI issue has be resolved.

Since the channel SNRs of these parallel channels are not the same, wecan apply the optimal Water filling solution to compute the transmitpower on each channel given a fixed total transmit power. Using thistransmit power and corresponding channel SNR, we can compute capacity ofthe equivalent system as given in the previous report.

Issues of Fading, Multipath, and Multi-Cell Interference

Techniques used to counteract channel fading (e.g., diversitytechniques) in conventional systems can also be applied in MMLO. Forslowly-varying multi-path dispersive channels, if the channel impulseresponse can be fed back, it can be incorporated into the equivalentsystem mentioned above, by which the channel induced ISI and theintentionally introduced MMLO ISI can be addressed jointly. For fasttime-varying channels or when channel feedback is impossible, channelequalization needs to be performed at the receiver. A block-basedfrequency-domain equalization can be applied and an oversampling wouldbe required.

If we consider the same adjacent channel power leakage for MMLO and theconventional system, then the adjacent cells' interference power wouldbe approximately the same for both systems. If interference cancellationtechniques are necessary, they can also be developed for MMLO.

Channel fading can be another source of intersymbol interference (ISI)and interlayer interference (ILI). One manner for representingsmall-scale signal fading is the use of statistical models. WhiteGaussian noise may be used to model system noise. The effects ofmultipath fading may be modeled using Rayleigh or Rician probabilitydensity functions. Additive white Gaussian noise (AWGN) may berepresented in the following manner. A received signal is:

r(t)=s(t)+n(t)

where: r(t)=a received signal; s(t)=a transmitted signal; andn(t)=random noise signal

Rayleigh fading functions are useful for predicting bit error rate (BER)any multipath environment. When there is no line of sight (LOS) ordominate received signal, the power the transmitted signal may berepresented by:

${P_{r}(r)} = \left\{ \begin{matrix}{{\frac{r}{\sigma^{2}}e^{\frac{- r^{2}}{2\; \sigma^{2}}}},} & {r \geq 0} \\{0,} & {r < 0}\end{matrix} \right.$

where: σ=rms value of received signal before envelope detection,

-   -   σ=time average power of the received signal before envelope        detection.

In a similar manner, Rician functions may be used in situations wherethere is a line of sight or dominant signal within a transmitted signal.In this case, the power of the transmitted signal can be represented by:

${P_{r}(r)} = \left\{ \begin{matrix}{{\frac{r}{\sigma^{2}}e^{\frac{- {({r^{2} + A^{2}})}}{2\; \sigma^{2}}}{{II}_{0}\left( \frac{A_{r}}{\sigma^{2}} \right)}},} & {A \geq r \geq 0} \\{0,} & {r < 0}\end{matrix} \right.$

where A=peak amplitude of LOS component

II₀=modified Bessel Function of the first kind and zero-order

These functions may be implemented in a channel simulation to calculatefading within a particular channel using a channel simulator such asthat illustrated in FIG. 63. The channel simulator 6302 includes aBernoulli binary generator 6304 for generating an input signal that isprovided to a rectangular M-QAM modulator 6306 that generates a QAMsignal at baseband. Multipath fading channel block 6308 uses the Ricianequations to simulate multipath channel fading. The simulated multipathfading channel is provided to a noise channel simulator 6310. The noisechannel simulator 6310 simulates AWGN noise. The multipath fadingchannel simulator 6308 further provides channel state information toarithmetic processing block 6312 which utilizes the simulated multipathfading information and the AWGN information into a signal that isdemodulated at QAM demodulator block 6314. The demodulated simulatedsignal is provided to the doubler block 6316 which is input to a receiveinput of an error rate calculator 6318. The error rate calculator 6318further receives at a transmitter input, the simulated transmissionsignal from the Bernoulli binary generator 6304. The error ratecalculator 6318 uses the transmitter input and the received input toprovide in error rate calculation to a bit error rate block 6320 thatdetermines the channel bit error rate. This type of channel simulationfor determining bit error rate will enable a determination of the amountof QLO that may be applied to a signal in order to increase throughputwithout overly increasing the bit error rate within the channel.

Scope and System Description

This report presents the symbol error probability (or symbol error rate)performance of MLO signals in additive white Gaussian noise channel withvarious inter-symbol interference levels. As a reference, theperformance of the conventional QAM without ISI is also included. Thesame QAM size is considered for all layers of MLO and the conventionalQAM.

The MLO signals are generated from the Physicist's special Hermitefunctions:

${f_{n}\left( {t,\alpha} \right)} = {\sqrt{\frac{\alpha}{\sqrt{\pi}{n!}2^{n}}}{H_{n}\left( {\alpha \; t} \right)}e^{- \frac{\alpha^{2}t^{2}}{2}}}$

where Hn(αt) is the n^(th) order Hermite polynomial. Note that thefunctions used in the lab setup correspond to

$\alpha = \frac{1}{\sqrt{2}}$

and, for consistency,

$\alpha = \frac{1}{\sqrt{2}}$

is used in this report.

MLO signals with 3, 4 or 10 layers corresponding to n=0˜2, 0˜3, or 0˜9are used and the pulse duration (the range of t) is [−8, 8] in the abovefunction.

AWGN channel with perfect synchronization is considered.

The receiver consists of matched filters and conventional detectorswithout any interference cancellation, i.e., QAM slicing at the matchedfilter outputs.

${\% \mspace{14mu} {pulse}\text{-}{overlapping}} = {\frac{T_{p} - T_{sym}}{T_{p}} \times 100\%}$

where Tp is the pulse duration (16 in the considered setup) and Tsym isthe reciprocal of the symbol rate in each MLO layer. The consideredcases are listed in the following table.

TABLE 7 % of Pulse Overlapping T_(sym) T_(p)   0% 16 16 12.5% 14 1618.75%  13 16   25% 12 16 37.5% 10 16 43.75%  9 16   50% 8 16 56.25%  716 62.5% 6 16   75% 4 16

Derivation of the Signals Used in Modulation

To do that, it would be convenient to express signal amplitude s(t) in acomplex form close to quantum mechanical formalism. Therefore thecomplex signal can be represented as:

ψ(t) = s(t) + j σ(t) where   s(t) ≡ real  signalσ(t) = imaginary  signal(quadrature)${\sigma (t)} = {\frac{1}{\pi}{\int_{- \infty}^{\infty}{{s(\tau)}\frac{d\; \tau}{\tau - t}}}}$${s(t)} = {{- \frac{1}{\pi}}{\int_{- \infty}^{\infty}{{\sigma (\tau)}\frac{d\; \tau}{\tau - t}}}}$

Where s(t) and σ(t) are Hilbert transforms of one another and since σ(t)is qudratures of s(t), they have similar spectral components. That is ifthey were the amplitudes of sound waves, the ear could not distinguishone form from the other.

Let us also define the Fourier transform pairs as follows:

${\psi (t)} = {\frac{1}{\pi}{\int_{- \infty}^{\infty}{{\phi (f)}e^{j\; \omega \; t}{df}}}}$${\phi (f)} = {\frac{1}{\pi}{\int_{- \infty}^{\infty}{{\psi (t)}e^{{- j}\; \omega \; t}{dt}}}}$ψ^(*)(t)ψ(t) = [s(t)]² + [σ(t)]² + …   ≡ signal  power

Let's also normalize all moments to M₀:

M₀ = ∫₀^(τ)s(t)dt  M₀ = ∫₀^(τ)ϕ^(*)ϕ df

Then the moments are as follows:

M₀ = ∫₀^(τ)s(t)dt M₁ = ∫₀^(τ)ts(t)dt M₂ = ∫₀^(τ)t²s(t)dtM_(N − 1) = ∫₀^(τ)t^(N − 1)s(t)dt

In general, one can consider the signal s(t) be represented by apolynomial of order N, to fit closely to s(t) and use the coefficient ofthe polynomial as representation of data. This is equivalent tospecifying the polynomial in such a way that its first N “moments” M_(j)shall represent the data. That is, instead of the coefficient of thepolynomial, we can use the moments. Another method is to expand thesignal s(t) in terms of a set of N orthogonal functions φ_(k)(t),instead of powers of time. Here, we can consider the data to be thecoefficients of the orthogonal expansion. One class of such orthogonalfunctions are sine and cosine functions (like in Fourier series).

Therefore we can now represent the above moments using the orthogonalfunction w with the following moments:

$\begin{matrix}{\overset{\_}{t} = \frac{\int{{\psi^{*}(t)}t\; {\psi (t)}{dt}}}{\int{{\psi^{*}(t)}{\psi (t)}{dt}}}} & {\overset{\_}{t^{2}} = \frac{\int{{\psi^{*}(t)}{t\;}^{2}{\psi (t)}{dt}}}{\int{{\psi^{*}(t)}\; {\psi (t)}{dt}}}} & \; \\{\overset{\_}{t^{n}} = \frac{\int{{\psi^{*}(t)}{t\;}^{n}{\psi (t)}{dt}}}{\int{{\psi^{*}(t)}{\psi (t)}{dt}}}} & \; & \;\end{matrix}$

Similarly,

$\begin{matrix}{\overset{\_}{f} = \frac{\int{{\phi^{*}(f)}f\; {\phi (f)}{df}}}{\int{{\phi^{*}(f)}{\phi (f)}{df}}}} & {\overset{\_}{f^{2}} = \frac{\int{{\phi^{*}(f)}f^{2}{\phi (f)}{df}}}{\int{{\phi^{*}(f)}{\phi (f)}{df}}}} \\{\overset{\_}{f^{n}} = \frac{\int{{\phi^{*}(f)}f^{n}{\phi (f)}{df}}}{\int{{\phi (f)}{\phi (f)}{df}}}} & \;\end{matrix}$

If we did not use complex signal, then:

f=0

To represent the mean values from time to frequency domains, replace:

ϕ(f) → ψ(t)$\left. f\rightarrow{\frac{1}{2\pi \; j}\frac{d}{dt}} \right.$

These are equivalent to somewhat mysterious rule in quantum mechanicswhere classical momentum becomes an operator:

$\left. P_{x}\rightarrow{\frac{h}{2\pi \; j}\frac{\partial}{\partial x}} \right.$

Therefore using the above substitutions, we have:

$\overset{\_}{f} = {\frac{\int{{\phi^{*}(f)}f\; \phi \; (f){df}}}{\int{{\phi^{*}(f)}\; \phi \; (f){df}}} = {\frac{\int{{\psi^{*}(t)}\left( \frac{1}{2\pi \; j} \right)\frac{d\; \psi \; (t)}{dt}{dt}}}{\int{{\psi^{*}(t)}{\psi (t)}{dt}}} = {\left( \frac{1}{2\pi \; j} \right)\frac{\int{\psi^{*}\frac{d\; \psi}{dt}{dt}}}{\int{\psi^{*}\psi \; {dt}}}}}}$

And:

$\begin{matrix}{\overset{\_}{f^{2}} = \frac{\int{{\phi^{*}(f)}f^{2}{\phi (f)}{df}}}{\int{{\phi^{*}(f)}{\phi (f)}{df}}}} & {= {\frac{\int{{\psi^{*}\left( \frac{1}{2\pi \; j} \right)}^{2}\frac{d^{2}}{{dt}^{2}}\; \psi \; {dt}}}{\int{\psi^{*}\psi \; {dt}}} = {{\quad\quad} - {\left( \frac{1}{2\pi} \right)^{2}\frac{\int{{\psi^{*}\left( \frac{d^{2}\;}{{dt}^{2}} \right)}\psi \; {dt}}}{\int{\psi^{*}\psi \; {dt}}}}}}} \\{\overset{\_}{t^{2}} = \frac{\int{\psi^{*}t^{2}\; \psi \; {dt}}}{\int{\psi^{*}\psi \; {dt}}}} & \;\end{matrix}$

We can now define an effective duration and effective bandwidth as:

Δt=√{square root over (2π(t−t)² )}=2π·rms in time

Δf=√{square root over (2π(f−f)² )}=2π·rms in frequency

But we know that:

(t−t)² = t ² −( t )²

(f−f)² = f ² −( f )²

We can simplify if we make the following substitutions:

τ=t−t

Ψ(τ)=ψ(t)e ^(−jωτ)

ω₀=ω=2πf=2πf ₀

We also know that:

(Δt)²(Δf)²=(Δt Δf)²

And therefore:

$\left( {\Delta \; t\; \Delta \; f} \right)^{2} = {{\frac{1}{4}\left\lbrack {4\frac{\int{{\Psi^{*}(\tau)}\tau^{2}{\Psi (\tau)}d\; \tau {\int{\frac{d\; \Psi^{*}}{d\; \tau}\frac{d\; \Psi}{d\; \tau}d\; \tau}}}}{\left( {\int{{\Psi^{*}(\tau)}{\psi (\tau)}d\; \tau}} \right)^{2}}} \right\rbrack} \geq \left( \frac{1}{4} \right)}$$\left( {\Delta \; t\; \Delta \; f} \right) \geq \left( \frac{1}{2} \right)$

Now instead of

$\left( {\Delta \; t\; \Delta \; f} \right) \geq \left( \frac{1}{2} \right)$

we are interested to force the equality

$\left( {\Delta \; t\; \Delta \; f} \right) = \left( \frac{1}{2} \right)$

and see what signals satisfy the equality. Given the fixed bandwidth Δf,the most efficient transmission is one that minimizes the time-bandwidthproduct

$\left( {\Delta \; t\; \Delta \; f} \right) = \left( \frac{1}{2} \right)$

For a given bandwidth Δf, the signal that minimizes the transmission inminimum time will be a Gaussian envelope. However, we are often givennot the effective bandwidth, but always the total bandwidth f₂−f₁. Now,what is the signal shape which can be transmitted through this channelin the shortest effective time and what is the effective duration?

$\left. {{\Delta \; t}==\frac{{\frac{1}{\left( {2\pi} \right)^{2}}{\int_{f_{1}}^{f_{2}}{\frac{d\; \phi^{\star}}{df}\frac{d\; \phi}{df}}}}\ }{\int_{f_{1}}^{f_{2}}{\phi^{\star}\phi \; {df}}}}\rightarrow\min \right.$

Where φ(f) is zero outside the range f₂−f₁.

To do the minimization, we would use the calculus of variations(Lagrange's Multiplier technique). Note that the denominator is constantand therefore we only need to minimize the numerator as:

$\mspace{79mu} {\left. {\Delta \; t}\rightarrow\left. \min\rightarrow{\delta {\int_{f_{1}}^{f_{2}}{\left( {{\frac{d\; \phi^{\star}}{df}\frac{d\; \phi}{df}} + {{\Lambda\phi}^{\star}\phi}} \right){df}}}} \right. \right. = 0}\ $First  Trem${\delta {\int_{f_{1}}^{f_{2}}{\frac{d\; \phi^{\star}}{df}\ \frac{d\; \phi}{df}{df}}}} = {{\int{\left( {{\frac{d\; \phi^{\star}}{df}\delta \frac{d\; \phi}{df}} + {\frac{d\; \phi}{df}\delta \frac{d\; \phi^{\star}}{df}}} \right){df}}} = \ {{\int{\left( {{\frac{d\; \phi^{\star}}{df}\frac{d\; {\delta\phi}}{df}} + {\frac{d\; \phi}{df}\frac{d\; \delta \; \phi^{\star}}{df}}} \right){df}}} = {{\left\lbrack {{\frac{d\; \phi^{\star}}{df}{\delta\phi}} + {\frac{d\; \phi}{df}{\delta\phi}^{\star}}} \right\rbrack_{f_{1}}^{f_{2}} - {\int{\left( {{\frac{d^{\; 2}\phi^{\star}}{{df}^{\mspace{11mu} 2}}{\delta\phi}} + {\frac{{d\;}^{2}\; \phi}{{df}^{\mspace{11mu} 2}}{\delta\phi}^{\star}}} \right){df}}}} = {\int{\left( {{\frac{d^{\; 2}\phi^{\star}}{{df}^{\mspace{11mu} 2}}{\delta\phi}} + {\frac{d^{\; 2}\; \phi}{{df}^{\mspace{11mu} 2}}{\delta\phi}^{\star}}} \right){df}}}}}}$     Second  Trem     δ∫_(f₁)^(f₂)(Λϕ^(⋆)ϕ)df = Λ∫_(f₁)^(f₂)(ϕ^(⋆)δϕ + ϕδϕ^(⋆))df$\mspace{79mu} {{{Both}\mspace{14mu} {Trems}}\mspace{79mu} = {{\int{\left\lbrack {{\left( {\frac{d^{\; 2}\phi^{\star}}{{df}^{\mspace{11mu} 2}} + {\Lambda\phi}^{\star}} \right){\delta\phi}} + {\left( {\frac{d^{\; 2}\phi}{{df}^{\mspace{11mu} 2}} + {\Lambda\phi}} \right){\delta\phi}^{\star}}} \right\rbrack {df}}} = 0}}$

This is only possible if and only if:

$\left( {\frac{d^{2}\phi}{{df}^{\; 2}} + {\Lambda\phi}} \right) = 0$

The solution to this is of the form

${\phi (f)} = {\sin \; k\; {\pi \left( \frac{f - f_{1}}{f_{2} - f_{1}} \right)}}$

Now if we require that the wave vanishes at infinity, but still satisfythe minimum time-bandwidth product:

$\left( {\Delta \; t\; \Delta \; f} \right) = \left( \frac{1}{2} \right)$

Then we have the wave equation of a Harmonic Oscillator:

${\frac{d^{2}{\Psi (\tau)}}{d\; \tau^{2}} + {\left( {\lambda - {\alpha^{2}\tau^{2}}} \right){\Psi (\tau)}}} = 0$

which vanishes at infinity only if:

λ = α(2n + 1)$\psi_{n} = {{e^{{- \frac{1}{2}}\omega^{2}\tau^{2}}\frac{d^{n}}{d\; \tau^{n}}e^{{- \alpha^{2}}\tau^{2}}} \propto {H_{n}(\tau)}}$

Where H_(n)(τ) is the Hermit functions and:

(Δt Δf)=½(2n+1)

So Hermit functions H_(n)(τ) occupy information blocks of 1/2, 3/2, 5/2,. . . with 1/2 as the minimum information quanta.

Squeezed States

Here we would derive the complete Eigen functions in the mostgeneralized form using quantum mechanical approach of Dirac algebra. Westart by defining the following operators:

$b = {\sqrt{\frac{m\; \omega^{\prime}}{2\hslash}}\left( {x + \frac{ip}{m\; \omega^{\prime}}} \right)}$$b^{+} = {{\sqrt{\frac{m\; \omega^{\prime}}{2\hslash}}{\left( {x - \frac{ip}{m\; \omega^{\prime}}} \right)\left\lbrack {b,b^{+}} \right\rbrack}} = 1}$a = λ b − μ b⁺ a⁺ = λ b⁺ − μ b

Now we are ready to define Δx and Δp as:

$\left( {\Delta \; x} \right)^{2} = {{\frac{\hslash}{2m\; \omega}\left( \frac{\omega}{\omega^{\prime}} \right)} = {\frac{\hslash}{2m\; \omega}\left( {\lambda - \mu} \right)^{2}}}$$\left( {\Delta \; p} \right)^{2} = {{\frac{\hslash \; m\; \omega}{2}\left( \frac{\omega^{\prime}}{\omega} \right)} = {\frac{\hslash \; m\; \omega}{2}\left( {\lambda + \mu} \right)^{2}}}$${\left( {\Delta \; x} \right)^{2}\left( {\Delta \; p} \right)^{2}} = {\frac{\hslash^{2}}{4}\left( {\lambda^{2} - \mu^{2}} \right)^{2}}$${\Delta \; x\; \Delta \; p} = {{\frac{\hslash}{2}\left( {\lambda^{2} - \mu^{2}} \right)} = \frac{\hslash}{2}}$

Now let parameterize differently and instead of two variables λ and μ,we would use only one variable ξ as follows:

λ=sin ξμ=cos ξλ+μ=e^(ξ)λ−μ=−e^(−ξ)

Now the Eigen states of the squeezed case are:

bβ⟩ = ββ⟩(λ a + μ a⁺)β⟩ = ββ⟩ b = UaU⁺U = e^(ξ/2(a² − a^(+²)))U⁺(ξ)aU(ξ) = a  cosh   ξ − a⁺sinh   ξU⁺(ξ)a⁺U(ξ) = a⁺cosh   ξ − a  sinh   ξ

We can now consider the squeezed operator:

${{{{{{{{{{{{{\alpha,\xi}\rangle} = {{U(\xi)}{D(\alpha)}}}}0}\rangle}{{D(\alpha)} = {e^{\frac{- {\alpha }^{2}}{2}}e^{\alpha \; a^{+}}e^{{- \alpha^{\star}}a}}}{{\alpha\rangle} = {\sum\limits_{n = 0}^{\infty}{\frac{\alpha^{n}}{\sqrt{n!}}e^{\frac{- {\alpha }^{2}}{2}}}}}}}n}\rangle}{{\alpha\rangle} = e^{\frac{- {\alpha }^{2}}{2} + {\alpha \; a^{+}}}}}}0}\rangle$

For a distribution P(n) we would have:

${P(n)} = {{{\langle{n{{\beta,\xi}\rangle}}}^{2}{\langle{{\alpha {{\beta,\xi}\rangle}} = {\sum\limits_{n = 0}^{\infty}{\frac{\alpha^{n}}{\sqrt{n!}}e^{\frac{- {\alpha }^{2}}{2}}{\langle{{n{{\beta,\xi}\rangle}e^{{2{zt}} - t^{2}}} = {\sum\limits_{n = 0}^{\infty}\frac{{H_{n}(z)}t^{n}}{n!}}}}}}}}}}$

Therefore the final result is:

$\langle{{n{{\beta,\xi}\rangle}} = {\frac{\left( {\tanh \mspace{11mu} \xi} \right)^{n/2}}{2^{n/2}\left( {{n!}\cosh \mspace{11mu} \xi} \right)^{2}}e^{{{- 1}/2}{({{\beta }^{2} - {\beta^{2}\tanh \; \xi}})}}H_{n}\mspace{11mu} \left( \frac{\beta}{2\mspace{11mu} \sinh \mspace{11mu} \xi \mspace{11mu} \cosh \mspace{11mu} \xi} \right)}}$

Another issue of concern with the use of QLO with QAM is a desire toimprove bit error rate (BER) performance without impacting theinformation rate or bandwidth requirements of the queue a low signal.One manner for improving BER performance utilizes two separateoscillators that are separated by a known frequency Δf. Signalsgenerated in this fashion will enable a determination of the BER.Referring now to FIG. 64, there is illustrated the generation of two bitstreams b1 and B2 that are provided to a pair of QAM modulators 6402 and6404 by a transmitter 6400. Modulator 6402 receives a first carrierfrequency F1 and modulator 6404 receives a second carrier frequency F2.The frequencies F1 and at two are separated by a known value Δf. Thesignals for each modulator are generated and combined at a summingcircuit 6406 to provide the output s(t). The variables in the outputs ofthe QAM modulators are A_(i) (amplitude), f_(i) (frequency) and φ_(i)(phase).

Therefore, each constituent QAM modulation occupies a bandwidth:

${BW} = {r_{s} = {\frac{r_{b}}{\log_{\; 2}\; m}\mspace{14mu} {{symbols}/\sec}}}$

where r_(s) equals the symbol rate of each constituent QAM signal.

The total bandwidth of signal s(t) is:

$W = {{r_{s}\left( {1 + \frac{\Delta \; f}{r_{s}}} \right)} = {r_{s} + {\Delta \; f\mspace{11mu} H_{z}}}}$

Therefore, the spectral efficiency η of this two oscillator system is:

$\eta = \frac{2r_{b}}{W}$ but r_(b) = r₂  log_( 2)  m$\eta = {\frac{2r_{b}}{W} = {\frac{2r_{s}\mspace{11mu} \log_{\; 2}\mspace{11mu} m}{r_{s}\left( {1 + \frac{\Delta \; f}{r_{s}}} \right)} = {\frac{2\mspace{11mu} \log_{\; 2}\mspace{11mu} m}{1 + \frac{\Delta \; f}{r_{s}}}\mspace{14mu} {{{bits}/\sec}/{Hz}}}}}$

The narrowband noise over the signal s(t) is:

n(t)=n _(I)(t)cos(2πf ₀ t)−n _(q)(t)sin(2πf ₀ t)

Where: n_(I)(t)=noise in I

N_(q)(t)=noise in Q

Each noise occupies a bandwidth of W [Hz] and the average power of eachcomponent is N₀W. N₀ is the noise power spectral density in Watts/Hz.The value of f₀ is the mean value of f₁ and f₂.

Referring now to FIG. 65, there is illustrated a receiver side blockdiagram for demodulating the signal generated with respect to FIG. 65.The received signal s(t)+n(t) is provided to a number of cosine filters6502-6508. Cosine filters 6502 and 6504 filter with respect to carrierfrequency f₁ and cosine filters 6506 and 6508 filter the received signalfor carrier frequency f₂. Each of the filters 6502-6508 provide anoutput to a switch 6510 that provides a number of output to atransformation block 6512. Transformation block 6512 provides two outputsignals having a real portion and an imaginary portion. Each of the realand imaginary portions associated with a signal are provided to anassociated decoding circuit 6514, 6516 to provide the decoded signals b₁and b₂.

[ a   ( T s ) b   ( T s ) c   ( T s ) d   ( T s ) ] = T s  [ 10 K 1 K 2 0 1 - K 2 K 1 K 1 - K 2 1 0 K 2 K 1 0 1 ]  [ A 1   ( cos  ϕ 1 ) A 1   ( sin   ϕ 1 ) A 2   ( cos   ϕ 2 ) A 2   ( sin  ϕ 2 ) ] + [ N I   1   ( T s ) N Q   1   ( T s ) N I   2  ( T s ) N Q   2   ( T s ) ]   | A >   ( nonsingular   so  it   has   - 1 )  | S >  | N >  | A >= T s   | S > + | N > Where${N_{I_{2 +}^{1 -}}\left( T_{s} \right)} = {{{\int_{0}^{T_{s}}{{\eta_{s}(t)}\cos \mspace{11mu} \left( {\frac{2{\eta\Delta}\; f}{2}t} \right)}} \mp {{\eta_{G}(t)}\sin \mspace{11mu} \left( {\frac{2{\eta\Delta}\; f}{2}t} \right){dt}{N_{Q_{2 +}^{1 -}}\left( T_{s} \right)}}} = \left. {{\int_{0}^{T_{s}}{{\eta_{I}(t)}{din}\mspace{11mu} \left( {\frac{2{\eta\Delta}\; f}{2}t} \right)}} \mp {{\eta_{Q}(t)}\cos \mspace{11mu} \left( {\frac{2{\eta\Delta}\; f}{2}t} \right){dt}}} \middle| {A>={T_{s}\; }} \middle| {S > +} \middle| {N >} \right.}$

Multiply by

$\frac{1}{T_{s}}$

⁻¹

1 T s   - 1 | A >= | S > + 1 T s   - 1 | N >= | S > + | N ~$\left. {Output}\mspace{14mu} \middle| > \mspace{104mu} \middle| {\overset{\sim}{N} > \begin{bmatrix}{I_{1}\mspace{11mu} \left( T_{s} \right)} \\{Q_{1}\mspace{11mu} \left( T_{s} \right)} \\{I_{2}\mspace{11mu} \left( T_{s} \right)} \\{Q_{2}\mspace{11mu} \left( T_{s} \right)}\end{bmatrix}} \right. = \left. {\begin{bmatrix}{A_{1}\mspace{11mu} \left( {\cos \mspace{11mu} \phi_{1}} \right)} \\{A_{1}\mspace{11mu} \left( {\sin \mspace{11mu} \phi_{1}} \right)} \\{A_{2}\mspace{11mu} \left( {\cos \mspace{11mu} \phi_{2}} \right)} \\{A_{2}\mspace{11mu} \left( {\sin \mspace{11mu} \phi_{2}} \right)}\end{bmatrix} + \begin{bmatrix}{{\overset{\sim}{N}}_{I\; 1}\mspace{11mu} \left( T_{s} \right)} \\{{\overset{\sim}{N}}_{Q\; 1}\mspace{11mu} \left( T_{s} \right)} \\{{\overset{\sim}{N}}_{I\; 2}\mspace{11mu} \left( T_{s} \right)} \\{{\overset{\sim}{N}}_{Q\; 2}\mspace{11mu} \left( T_{s} \right)}\end{bmatrix}}\mspace{31mu} \middle| >  \middle| {S >}\mspace{115mu} \middle| {\overset{\sim}{N} >} \right.$

Then the probability of correct decision P_(e) is

P _(e)≈(1−P _(e))⁴≈1−4P _(e) for P _(e)<<1

P_(e)=well known error probability in one dimension for each consistuentm-QAM modulation. Therefore, one can calculate BER.

P_(e) comprises the known error probability in one dimension for eachconstituent member of the QAM modulation. Using the known probabilityerror the bit error rate for the channel based upon the known differencebetween frequencies f₁ and f₂ may be calculated.

Adaptive Processing

The processing of signals using QLO may also be adaptively selected tocombat channel impairments and interference. The process for adaptiveQLO is generally illustrated in FIG. 66. First at step 6602 an analysisof the channel environment is made to determine the present operatingenvironment. The level of QLO processing is selected at step 6604 basedon the analysis and used to conFig. communications. Next, at step 6606,the signals are transmitted at the selected level of QLO processing.Inquiry step 6608 determines if sufficient channel quality has beenachieved. If so, the system continues to transmit and the selected QLOprocessing level at step 6606. If not, control passes back to step 6602to adjust the level of QLO processing to achieve better channelperformance.

The processing of signals using mode division multiplexing (MDM) mayalso be adaptively selected to combat channel impairments andinterference and maximize spectral efficiency. The process for adaptiveMDM is generally illustrated in FIG. 67. First at step 6702 an analysisof the channel environment is made to determine the present operatingenvironment. The level of MDM processing is selected at step 6704 basedon the analysis and used to conFig. communications. Next, at step 6706,the signals are transmitted at the selected level of MDM processing.Inquiry step 6708 determines if sufficient channel quality has beenachieved. If so, the system continues to transmit and the selected MDMprocessing level at step 6706. If not, control passes back to step 6702to adjust the level of MDM processing to achieve better channelperformance.

The processing of signals using an optimal combination of QLO and MDMmay also be adaptively selected to combat channel impairments andinterference and maximize spectral efficiency. The process for adaptiveQLO and MDM is generally illustrated in FIG. 68. First at step 6802 ananalysis of the channel environment is made to determine the presentoperating environment. A selected combination of a level of QLO processand a level of MDM processing are selected at step 6804 based on theanalysis and used to conFig. communications. Next, at step 6806, thesignals are transmitted at the selected level of QLO and MDM processing.Inquiry step 6808 determines if sufficient channel quality has beenachieved. If so, the system continues to transmit and the selectedcombination of QLO and MDM processing levels at step 6806. If not,control passes back to step 6802 to adjust the levels of QLO and MDMprocessing to achieve better channel performance. Adjustments throughthe steps continue until a most optimal combination of QLO and MDMprocessing is achieved to maximize spectral efficiency using a2-dimensional optimization.

The processing of signals using an optimal combination of QLO and QAMmay also be adaptively selected to combat channel impairments andinterference and maximize spectral efficiency. The process for adaptiveQLO and QAM is generally illustrated in FIG. 69. First at step 6902 ananalysis of the channel environment is made to determine the presentoperating environment. A selected combination of a level of QLO processand a level of QAM processing are selected at step 6904 based on theanalysis and used to conFig. communications. Next, at step 6906, thesignals are transmitted at the selected level of QLO and QAM processing.Inquiry step 6908 determines if sufficient channel quality has beenachieved. If so, the system continues to transmit and the selectedcombination of QLO and QAM processing levels at step 6906. If not,control passes back to step 6902 to adjust the levels of QLO and QAMprocessing to achieve better channel performance. Adjustments throughthe steps continue until a most optimal combination of QLO and QAMprocessing is achieved to maximize spectral efficiency using a2-dimensional optimization.

The processing of signals using an optimal combination of QLO, MDM andQAM may also be adaptively selected to combat channel impairments andinterference and maximize spectral efficiency. The process for adaptiveQLO, MDM and QAM is generally illustrated in FIG. 70. First at step 7002an analysis of the channel environment is made to determine the presentoperating environment. A selected combination of a level of QLOprocessing, a level of MDM processing and a level of QAM processing areselected at step 7004 based on the analysis and used to conFig.communications. Next, at step 7006, the signals are transmitted at theselected level of QLO, MDM and QAM processing. Inquiry step 7008determines if sufficient channel quality has been achieved. If so, thesystem continues to transmit and the selected combination of QLO, MDMand QAM processing levels at step 7006. If not, control passes back tostep 7002 to adjust the levels of QLO, MDM and QAM processing to achievebetter channel performance. Adjustments through the steps continue untila most optimal combination of QLO, MDM and QAM processing is achieved tomaximize spectral efficiency using a 3-dimensional optimization.

The adaptive approaches described herein above may be used with anycombination of QLO, MDM and QAM processing in order to achieve optimalchannel efficiency. In another application distinct modal combinationsmay also be utilized.

Improvement of Pilot Signal Modulation

The above described QLO, MDM and QAM processing techniques may also beused to improve the manner in which a system deals with noise, fadingand other channel impairments by the use of pilot signal modulationtechniques. As illustrated in FIG. 71, a pilot signal 7102 istransmitted between a transmitter 7104 to a receiver 7106. The pilotsignal includes an impulse signal that is received, detected andprocessed at the receiver 7106. Using the information received from thepilot impulse signal, the channel 7108 between the transmitter 7104 andreceiver 7106 may be processed to remove noise, fading and other channelimpairment issues from the channel 7108. The pilot signal assistedmodulation may also be carried out using orthogonal modes in anelliptical core fiber as discussed herein below.

This process is generally described with respect to the flowchart ofFIG. 72. The pilot impulse signal is transmitted at 7202 over thetransmission channel. The impulse response is detected at step 7204 andprocessed to determine the impulse response over the transmissionchannel. Effects of channel impairments such as noise and fading may becountered by multiplying signals transmitted over the transmissionchannel by the inverse of the impulse response at step 7206 in order tocorrect for the various channel impairments that may be up on thetransmission channel. In this way the channel impairments arecounteracted and improved signal quality and reception may be providedover the transmission channel.

Power Control

Adaptive power control may be provided on systems utilizing QLO, MDM andQAM processing to also improve channel transmission. Amplifiernonlinearities within the transmission circuitry and the receivercircuitry will cause impairments in the channel response as moreparticularly illustrated in FIG. 73. As can be seen the channelimpairments and frequency response increase and decrease over frequencyas illustrated generally at 7302. By adaptively controlling the power ofa transmitting unit or a receiving unit and inverse frequency responsesuch as that generated at 7304 may be generated. Thus, when the normalfrequency response 7302 and the inverse frequency response 7304 arecombined, a consistent response 7306 is provided by use of the adaptivepower control.

Backward and Forward Channel Estimation

QLO techniques may also be used with forward and backward channelestimation processes when communications between a transmitter 7402 anda receiver 7404 do not have the same channel response over both theforward and backward channels. As shown in FIG. 74, the forward channel7406 and backward channel 7408 between a transmitter 7402 and receiver7404 me each be processed to determine their channel impulse responses.Separate forward channel estimation response and backward channelestimation response may be used for processing QLO signals transmittedover the forward channel 7406 and backward channel 7408. The differencesin the channel response between the forward channel 7406 and thebackward channel 7408 may arise from differences in the topography ornumber of buildings located within the area of the transmitter 7402 andthe receiver 7404. By treating each of the forward channel 7406 and abackward channel 7408 differently better overall communications may beachieved.

Using MIMO Techniques with QLO

MIMO techniques may be used to improve the performance of QLO-basedtransmission systems. MIMO (multiple input and multiple output) is amethod for multiplying the capacity of a radio link using multipletransmit and receive antennas to exploit multipath propagation. MIMOuses multiple antennas to transmit a signal instead of only a singleantenna. The multiple antennas may transmit the same signal usingmodulation with the signals from each antenna modulated by differentorthogonal signals such as that described with respect to the QLOmodulation in order to provide an improved MIMO based system.

Diversions within OAM beams may also be reduced using phased arrays. Byusing multiple transmitting elements in a geometrical configuration andcontrolling the current and phase for each transmitting element, theelectrical size of the antenna increases as does the performance of theantenna. The antenna system created by two or more individual intendedelements is called an antenna array. Each transmitting element does nothave to be identical but for simplification reasons the elements areoften alike. To determine the properties of the electric field from anarray the array factor (AF) is utilized.

The total field from an array can be calculated by a superposition ofthe fields from each element. However, with many elements this procedureis very unpractical and time consuming. By using different kinds ofsymmetries and identical elements within an array, a much simplerexpression for the total field may be determined. This is achieved bycalculating the so-called array factor (AF) which depends on thedisplacement (and shape of the array), phase, current amplitude andnumber of elements. After calculating the array factor, the total fieldis obtained by the pattern multiplication rule which is such that thetotal field is the product of the array factor in the field from onesingle element.

E _(total) =E _(single element) ×AF

This formula is valid for all arrays consisting of identical elements.The array factor does not depend on the type of elements used, so forcalculating AF it is preferred to use point sources instead of theactual antennas. After calculating the AF, the equation above is used toobtain the total field. Arrays can be 1D (linear), 2D (planar) or 3D. Ina linear array, the elements are placed along the line and in a planarthey are situated in a plane.

Referring now to FIG. 75, there is illustrated in the manner in whichHermite Gaussian beams and Laguerre Gaussian beams will diverge whentransmitted from a phased array of antennas. For the generation ofLaguerre Gaussian beams a circular symmetry over the cross-section ofthe phased antenna array is used, and thus, a circular grid will beutilized. For the generation of Hermite Gaussian beams 7502, arectangular array 7504 of array elements 7506 is utilized. As can beseen with respect to FIG. 75, the Hermite Gaussian waves 7508 provide amore focused beam front then the Laguerre Gaussian waves 7510.

Reduced beam divergence may also be accomplished using a pair of lenses.As illustrated in FIG. 76A, a Gaussian wave 7602 passing through aspiral phase plate 7604 generates an output Laguerre Gaussian wave 7606.The Laguerre Gaussian wave 7606 when passing from a transmitter aperture7608 to a receiver aperture 7610 diverges such that the entire LaguerreGaussian beam does not intersect the receiver aperture 7610. This issuemay be addressed as illustrated in FIG. 76B. As before the Gaussianwaves 7602 pass through the spiral phase plate 7604 generating LaguerreGaussian waves 7606. Prior to passing through the transmitter aperture7608 the Laguerre Gaussian waves 7606 pass through a pair of lenses7614. The pair of lenses 7614 have an effective focal length 7616 thatfocuses the beam 7618 passing through the transmitter aperture 7608. Dueto the focusing lenses 7614, the focused beam 7618 fully intersects thereceiver aperture 7612. By providing the lenses 7614 separated by aneffective focal length 7616, a more focused beam 7618 may be provided atthe receiver aperture 7612 preventing the loss of data within thetransmission of the Laguerre Gaussian wave 7606.

Application of OAM to Optical Communication

Utilization of OAM for optical communications is based on the fact thatcoaxially propagating light beams with different OAM states can beefficiently separated. This is certainly true for orthogonal modes suchas the LG beam. Interestingly, it is also true for general OAM beamswith cylindrical symmetry by relying only on the azimuthal phase.Considering any two OAM beams with an azimuthal index of l 1 and l 2,respectively:

U ₁(r,θ,z)=A ₁(r,z)exp(il ₁θ)  (12)

where r and z refers to the radial position and propagation distancerespectively, one can quickly conclude that these two beams areorthogonal in the sense that:

$\begin{matrix}{{\int_{0}^{2\pi}{U_{1}U_{2}^{*}d\; \theta}} = \left\{ \begin{matrix}0 & {{{if}\mspace{14mu} _{1}} \neq _{2}} \\{A_{1}A_{2}^{*}} & {{{if}\mspace{14mu} _{1}} = _{2}}\end{matrix} \right.} & (13)\end{matrix}$

There are two different ways to take advantage of the distinctionbetween OAM beams with different l states in communications. In thefirst approach, N different OAM states can be encoded as N differentdata symbols representing “0”, “1”, . . . , “N−1”, respectively. Asequence of OAM states sent by the transmitter therefore represents datainformation. At the receiver, the data can be decoded by checking thereceived OAM state. This approach seems to be more favorable to thequantum communications community, since OAM could provide for theencoding of multiple bits (log 2(N)) per photon due to the infinitelycountable possibilities of the OAM states, and so could potentiallyachieve a higher photon efficiency. The encoding/decoding of OAM statescould also have some potential applications for on-chip interconnectionto increase computing speed or data capacity.

The second approach is to use each OAM beam as a different data carrierin an SDM (Spatial Division Multiplexing) system. For an SDM system, onecould use either a multi-core fiber/free space laser beam array so thatthe data channels in each core/laser beam are spatially separated, oruse a group of orthogonal mode sets to carry different data channels ina multi-mode fiber (MMF) or in free space. Greater than 1 petabit/s datatransmission in a multi-core fiber and up to 6 linearly polarized (LP)modes each with two polarizations in a single core multi-mode fiber hasbeen reported. Similar to the SDM using orthogonal modes, OAM beams withdifferent states can be spatially multiplexed and demultiplexed, therebyproviding independent data carriers in addition to wavelength andpolarization. Ideally, the orthogonality of OAM beams can be maintainedin transmission, which allows all the data channels to be separated andrecovered at the receiver. A typical embodiments of OAM multiplexing isconceptually depicted in FIG. 27. An obvious benefit of OAM multiplexingis the improvement in system spectral efficiency, since the samebandwidth can be reused for additional data channels.

Optical Fiber Communications

The use of orbital angular momentum and multiple layer overlaymodulation processing techniques within an optical communicationsinterface environment as described with respect to FIG. 3 can provide anumber of opportunities within the optical communications environmentfor enabling the use of the greater signal bandwidths provided by theuse of optical orbital angular momentum processing, or multiple layeroverlay modulation techniques alone. FIG. 77 illustrates the generalconfiguration of an optical fiber communication system. The opticalfiber communication system 7700 includes an optical transmitter 7702 andan optical receiver 7704. The transmitter 7702 and receiver 7704communicate over an optical fiber 7706. The transmitter 7702 includesinformation within a light wavelength or wavelengths that is propagatedover the optical fiber 7706 to the optical receiver 7704.

Optical communications network traffic has been steadily increasing by afactor of 100 every decade. The capacity of single mode optical fibershas increased 10,000 times within the last three decades. Historically,the growth in the bandwidth of optical fiber communications has beensustained by information multiplexing techniques using wavelength,amplitude, phase, and polarization of light as a means for encodinginformation. Several major discoveries within the fiber-optics domainhave enabled today's optical networks. An additional discovery was ledby Charles M. Kao's groundbreaking work that recognized glass impuritieswithin an optical fiber as a major signal loss mechanism. Existing glasslosses at the time of his discovery were approximately 200 dB perkilometer at 1 micrometer.

These discoveries gave birth to optical fibers and led to the firstcommercial optical fibers in the 1970s, having an attenuation low enoughfor communication purposes in the range of approximately 20 dBs perkilometer. Referring now to FIGS. 78a-78c , there is more particularlyillustrated the single mode fiber 7802, multicore fibers 7808, andmultimode fibers 7810 described herein above. The multicore fibers 7808consist of multiple cores 7812 included within the cladding 7813 of thefiber. As can be seen in FIG. 78b , there are illustrated a 3 corefiber, 7 core fiber, and 19 core fiber. Multimode fibers 7810 comprisemultimode fibers comprising a few mode fiber 7820 and a multimode fiber7822. Finally, there is illustrated a hollow core fiber 7815 including ahollow core 7814 within the center of the cladding 7816 and sheathing7818. The development of single mode fibers (SMF) such as thatillustrated at 7802 (FIG. 78a ) in the early 1980s reduced pulsedispersion and led to the first fiber-optic based trans-Atlantictelephone cable. This single mode fiber included a single transmissioncore 7804 within an outer sheathing 7806. Development of indium galliumarsenide photodiodes in the early 1990s shifted the focus tonear-infrared wavelengths (1550 NM), were silica had the lowest loss,enabling extended reach of the optical fibers. At roughly the same time,the invention of erbium-doped fiber amplifiers resulted in one of thebiggest leaps in fiber capacity within the history of communication, athousand fold increase in capacity occurred over a 10 year period. Thedevelopment was mainly due to the removed need for expensive repeatersfor signal regeneration, as well as efficient amplification of manywavelengths at the same time, enabling wave division multiplexing (WDM).

Throughout the 2000s, increases in bandwidth capacity came mainly fromintroduction of complex signal modulation formats and coherentdetection, allowing information encoding using the phase of light. Morerecently, polarization division multiplexing (PDM) doubled channelcapacity. Through fiber communication based on SMFs featured tremendousgrowth in the last three decades, recent research has indicated SMFlimitations. Non-linear effects in silica play a significant role inlong range transmission, mainly through the Kerr effect, where apresence of a channel at one wavelength can change the refractive indexof a fiber, causing distortions of other wavelength channels. Morerecently, a spectral efficiency (SE) or bandwidth efficiency, referringto the transmitted information rate over a given bandwidth, has becometheoretically analyzed assuming nonlinear effects in a noisy fiberchannel. This research indicates a specific spectral efficiency limitthat a fiber of a certain length can reach for any signal to noise(SNR). Recently achieved spectral efficiency results indeed show thatthe proximity to the spectral efficiency limit, indicating the need fornew technologies to address the capacity issue in the future.

Among several possible directions for optical communications in thefuture, the introduction of new optical fibers 7706 other than singlemode fibers 7802 has shown promising results. In particular, researchershave focused on spatial dimensions in new fibers, leading to so-calledspace division multiplexing (SDM) where information is transmitted usingcores of multi-core fibers (MCF) 7808 (FIG. 78b ) or mode divisionmultiplexing (MDM) or information is transmitted using modes ofmultimode fibers (MMFs) 7810 (FIG. 78c ). The latest results showspectral efficiency of 91 bits/S/Hz using 12 core multicore fiber 7808for 52 kilometer long fibers and 12 bits/S/Hz using 6 mode multimodefiber 7810 and 112 kilometer long fibers. Somewhat unconventionaltransmissions at 2.08 micrometers have also been demonstrated in two 90meter long photonic crystal fibers, though these fibers had high lossesof 4.5 decibels per kilometer.

While offering promising results, these new types of fibers have theirown limitations. Being noncircularly symmetric structures, multicorefibers are known to require more complex, expensive manufacturing. Onthe other hand, multimode fibers 7810 are easily created using existingtechnologies. However, conventional multimode fibers 7810 are known tosuffer from mode coupling caused by both random perturbations in thefibers and in modal multiplexers/demultiplexers.

Several techniques have been used for mitigating mode coupling. In astrong coupling regime, modal cross talk can be compensated usingcomputationally intensive multi-input multi-output (MIMO) digital signalprocessing (DSP). While MIMO DSP leverages the technique's currentsuccess in wireless networks, the wireless network data rates areseveral orders of magnitude lower than the ones required for opticalnetworks. Furthermore, MIMO DSP complexity inevitably increases with anincreasing number of modes and no MIMO based data transmissiondemonstrations have been demonstrated in real time thus far.Furthermore, unlike wireless communication systems, optical systems arefurther complicated because of fiber's nonlinear effects. In a weakcoupling regime, where cross talk is smaller, methods that also usecomputationally intensive adapted optics, feedback algorithms have beendemonstrated. These methods reverse the effects of mode coupling bysending a desired superposition of modes at the input, so that desiredoutput modes can be obtained. This approach is limited, however, sincemode coupling is a random process that can change on the order of amillisecond in conventional fibers.

Thus, the adaptation of multimode fibers 7810 can be problematic in longhaul systems where the round trip signal propagation delay can be tensof milliseconds. Though 2×56 GB/S transmission at 8 kilometers lengthhas been demonstrated in the case of two higher order modes, none of theadaptive optics MDM methods to date have demonstrated for more than twomodes. Optical fibers act as wave guides for the information carryinglight signals that are transmitted over the fiber. Within an ideal case,optical fibers are 2D, cylindrical wave guides comprising one or severalcores surrounded by a cladding having a slightly lower refractive indexas illustrated in FIGS. 78a-78d . A fiber mode is a solution (aneigenstate) of a wave guide equation describing the field distributionthat propagates within a fiber without changing except for the scalingfactor. All fibers have a limit on the number of modes that they canpropagate, and have both spatial and polarization degrees of freedom.

Single mode fibers (SMFs) 7802 is illustrated in FIG. 78a supportpropagation of two orthogonal polarizations of the fundamental mode only(N=2). For sufficiently large core radius and/or the core claddingdifference, a fiber is multimoded for N>2 as illustrated in FIG. 78c .For optical signals having orbital angular momentums and multilayermodulation schemes applied thereto, multimode fibers 7810 that areweakly guided may be used. Weakly guided fibers have a core claddingrefractive index difference that is very small. Most glass fibersmanufactured today are weakly guided, with the exception of somephotonic crystal fibers and air-core fibers. Fiber guide modes ofmultimode fibers 7810 may be associated in step indexed groups where,within each group, modes typically having similar effective indexes aregrouped together. Within a group, the modes are degenerate. However,these degeneracies can be broken in a certain fiber profile design.

We start by describing translationally invariant waveguide withrefractive index n=n(x, y), with n_(co) being maximum refractive index(“core” of a waveguide), and n_(cl) being refractive index of theuniform cladding, and p represents the maximum radius of the refractiveindex n. Due to translational invariance the solutions (or modes) forthis waveguide can be written as:

E _(j)(x,y,z)=e _(j)(x,y)e ^(iβ6) ^(j) ^(z),

H _(j)(x,y,z)=h _(j)(x,y)e ^(iβ) ^(j) ^(z),

where β_(j) is the propagation constant of the j-th mode. Vector waveequation for source free Maxwell's equation can be written in this caseas:

(∇² +n ² k ²−β_(j) ²)e _(j)=−(∇_(t) +iβ _(j) {circumflex over (z)})(e_(tj)·∇_(t) ln(n ²))

(∇² +n ² k ²−β_(j) ²)h _(j)=−(∇_(t) ln(n ²))×(

(∇)

_(t) +iβ _(j) {circumflex over (z)})×h _(j))

where k=2π/λ is the free-space wavenumber, λ is a free-space wavelength,e_(t)=e_(x){circumflex over (x)}+e_(y)ŷ is a transverse part of theelectric field, ∇² is a transverse Laplacian and ∇_(t) transverse vectorgradient operator. Waveguide polarization properties are built into thewave equation through the ∇_(t) ln(n²) terms and ignoring them wouldlead to the scalar wave equation, with linearly polarized modes. Whileprevious equations satisfy arbitrary waveguide profile n(x, y), in mostcases of interest, profile height parameter Δ can be considered smallΔ<<1, in which case waveguide is said to be weakly guided, or thatweakly guided approximation (WGA) holds. If this is the case, aperturbation theory can be applied to approximate the solutions as:

E(x,y,z)=e(x,y)e ^(i(β+{tilde over (β)})z)=(e _(t) +{circumflex over(z)}e _(z))e ^(i(β+{tilde over (β)}))z

H(x,y,z)=h(x,y)e ^(i(β+{tilde over (β)})z)=(h _(t) +{circumflex over(z)}h _(z))e ^(i(β+{tilde over (β)})z)

where subscripts t and z denote transverse and longitudinal componentsrespectively. Longitudinal components can be considered much smaller inWGA and we can approximate (but not neglect) them as:

$e_{z} = {\frac{{i\left( {2\Delta} \right)}^{\frac{1}{2}}}{v}\left( {\rho {\nabla_{t}{\cdot e_{t}}}} \right)}$$h_{z} = {\frac{{i\left( {2\Delta} \right)}^{\frac{1}{2}}}{V}\left( {\rho {\nabla_{t}{\cdot h_{t}}}} \right)}$

Where Δ and V are profile height and fiber parameters and transversalcomponents satisfy the simplified wave equation.

(∇² +n ² k ²−β_(j) ²)e _(j)=0

Though WGA simplified the waveguide equation, further simplification canbe obtained by assuming circularly symmetric waveguide (such as idealfiber). If this is the case refractive index that can be written as:

n(r)=n ² _(co)(1−2f(R)Δ)

where f(R)≧0 is a small arbitrary profile variation.

For a circularly symmetric waveguide, we would have propagationconstants β_(lm) that are classified using azimuthal (l) and radial (m)numbers. Another classification uses effective indices n_(lm) (sometimesnoted as n^(eff) _(lm) or simply n_(eff), that are related topropagation constant as: β_(lm)=kn^(ef f)). For the case of l=0, thesolutions can be separated into two classes that have either transverseelectric (T E_(0m)) or transverse magnetic (T M_(0m)) fields (calledmeridional modes). In the case of l≠0, both electric and magnetic fieldhave z-component, and depending on which one is more dominant, so-calledhybrid modes are denoted as: HE_(lm) and EH_(lm).

Polarization correction δβ has different values within the same group ofmodes with the same orbital number (l), even in the circularly symmetricfiber. This is an important observation that led to development of aspecial type of fiber.

In case of a step refractive index, solutions are the Bessel functionsof the first kind, J_(l)(r), in the core region, and modified Besselfunctions of the second kind, K_(l)(r), in the cladding region.

In the case of step-index fiber the groups of modes are almostdegenerate, also meaning that the polarization correction δβ can beconsidered very small. Unlike HE₁₁ modes, higher order modes (HOMs) canhave elaborate polarizations. In the case of circularly symmetric fiber,the odd and even modes (for example HE^(odd) and HE^(even) modes) arealways degenerate (i.e. have equal n_(eff)), regardless of the indexprofile. These modes will be non-degenerate only in the case ofcircularly asymmetric index profiles.

Referring now to FIG. 79, there are illustrated the first six modeswithin a step indexed fiber for the groups L=0 and L=1.

When orbital angular momentums are applied to the light wavelengthwithin an optical transmitter of an optical fiber communication system,the various orbital angular momentums applied to the light wavelengthmay transmit information and be determined within the fiber mode.

Angular momentum density (M) of light in a medium is defined as:

$M = {{\frac{1}{c^{2}}r \times \left( {E \times H} \right)} = {{r \times P} = {\frac{1}{c^{2}}r \times S}}}$

with r as position, E electric field, H magnetic field, P linearmomentum density and S Poynting vector.

The total angular momentum (J), and angular momentum flux (Ψ_(M)) can bedefined as:

J=∫∫∫M dV

Ψ=∫∫M dA

In order to verify whether certain mode has an OAM let us look at thetime averages of the angular momentum flux Ψ_(M):

Ψ_(M)

=∫∫

M

dA

as well as the time average of the energy flux:

${\langle\Phi_{W}\rangle} = {\int{\int{\frac{\langle S_{z}\rangle}{c}{dA}}}}$

Because of the symmetry of radial and axial components about the fiberaxis, we note that the integration in equation will leave onlyz-component of the angular momentum density non zero. Hence:

${\langle M\rangle} = {{\langle M\rangle}_{z} = {\frac{1}{c^{2}}r \times {\langle{E \times H}\rangle}_{z}}}$

and knowing (S)=Re{S} and S=½ E×H* leads to:

S _(Φ)=½(−E _(r) H _(z) *+E _(z) H _(r)*)

S _(z)=½(E _(x) H _(y) *−E _(y) H _(x)*)

Let us now focus on a specific linear combination of the HE_(l+1,m)^(even) and HE_(l+1,m) ^(odd) modes with π/2 phase shift among them:

V _(lm) ⁺ =HE _(l+1,m) ^(even) +iEH _(l+1,m) ^(odd)

The idea for this linear combination comes from observing azimuthaldependence of the HE_(l+1,m) ^(even) and modes comprising cos(φ) and sin(φ). If we denote the electric field of HEM_(l+1,m) ^(even) andHE_(l+1,m) ^(odd) modes as e₁ and e₂, respectively, and similarly,denote their magnetic fields as h₁ and h₂, the expression for theis newmode can be written as:

e=e ₁ +ie ₂,  (2.35)

h=h ₁ +ih ₂.  (2.36)

then we derive:

e_(r) = e^(i(l + 1)ϕ)F_(l)(R)$h_{z} = {e^{{i{({l + 1})}}\phi}{n_{co}\left( \frac{\varepsilon_{0}}{\mu_{0}} \right)}^{\frac{1}{2}}\frac{\left( {2\Delta} \right)^{\frac{1}{2}}}{V}G_{l}^{-}}$$e_{z} = {{ie}^{{i{({l + 1})}}\phi}\mspace{11mu} \frac{\left( {2\Delta} \right)^{\frac{1}{2}}}{V}G_{l}^{-}}$$h_{r} = {{- {ie}^{{i{({l + 1})}}\phi}}{n_{co}\left( \frac{\varepsilon_{0}}{\mu_{0}} \right)}^{\frac{1}{2}}{F_{l}(R)}}$

Where F_(l)(R) is the Bessel function and

$G_{l}^{\pm} = {\frac{{dF}_{l}}{dR} \pm {\frac{l}{R}F_{l}}}$

We note that all the quantities have e^(i(l+1)φ) dependence thatindicates these modes might have OAM, similarly to the free space case.Therefore the azimuthal and the longitudinal component of the Poyntingvector are:

$S_{\phi} = {{- {n_{co}\left( \frac{\varepsilon_{0}}{\mu_{0}} \right)}^{\frac{1}{2}}}\frac{\left( {2\Delta} \right)^{\frac{1}{2}}}{V}{Re}\left\{ {F_{l}^{*}G_{l}^{-}} \right\}}$$S_{z} = {{n_{co}\left( \frac{\varepsilon_{0}}{\mu_{0}} \right)}^{\frac{1}{2}}\left\lceil F_{l} \right\rceil^{2}}$

The ratio of the angular momentum flux to the energy flux thereforebecomes:

$\frac{\varphi_{M}}{\varphi_{W}} = \frac{l + 1}{\omega}$

We note that in the free-space case, this ratio is similar:

$\frac{\varnothing_{M}}{\varnothing_{W}} = \frac{\sigma + 1}{\omega}$

where σ represents the polarization of the beam and is bounded to be−1<σ<1. In our case, it can be easily shown that SAM of the V⁺ state, is1, leading to important conclusion that the OAM of the V^(+lm) state isl. Hence, this shows that, in an ideal fiber, OAM mode exists.

Thus, since an orbital angular momentum mode may be detected within theideal fiber, it is possible to encode information using this OAM mode inorder to transmit different types of information having differentorbital angular momentums within the same optical wavelength.

The above description with respect to optical fiber assumed an idealscenario of perfectly symmetrical fibers having no longitudinal changeswithin the fiber profile. Within real world fibers, random perturbationscan induce coupling between spatial and/or polarization modes, causingpropagating fields to evolve randomly through the fiber. The randomperturbations can be divided into two classes, as illustrated in FIG.80. Within the random perturbations 8002, the first class comprisesextrinsic perturbations 8004. Extrinsic perturbations 8004 includestatic and dynamic fluctuations throughout the longitudinal direction ofthe fiber, such as the density and concentration fluctuations natural torandom glassy polymer materials that are included within fibers. Thesecond class includes extrinsic variations 8006 such as microscopicrandom bends caused by stress, diameter variations, and fiber coredefects such as microvoids, cracks, or dust particles.

Mode coupling can be described by field coupling modes which account forcomplex valued modal electric field amplitudes, or by power couplingmodes, which is a simplified description that accounts only for realvalue modal powers. Early multimode fiber systems used incoherent lightemitting diode sources and power coupling models were widely used todescribe several properties including steady state, modal powerdistributions, and fiber impulse responses. While recent multimode fibersystems use coherent sources, power coupling modes are still used todescribe effects such as reduced differential group delays and plasticmultimode fibers.

By contrast, single mode fiber systems have been using laser sources.The study of random birefringence and mode coupling in single modefibers which leads to polarization mode dispersion (PMD), uses fieldcoupling modes which predict the existence of principal states ofpolarization (PSPs). PSPs are polarization states shown to undergominimal dispersion and are used for optical compensation of polarizationmode dispersion in direct detection single mode fiber systems. In recentyears, field coupling modes have been applied to multimode fibers,predicting principal mode which are the basis for optical compensationof modal dispersion in direct detection multimode fiber systems.

Mode coupling can be classified as weak or strong, depending on whetherthe total system length of the optical fiber is comparable to, or muchlonger than, a length scale over which propagating fields remaincorrelated. Depending on the detection format, communication systems canbe divided into direct and coherent detection systems. In directdetection systems, mode coupling must either be avoided by carefuldesign of fibers and modal D (multiplexers) and/or mitigated by adaptiveoptical signal processing. In systems using coherent detection, anylinear cross talk between modes can be compensated by multiple inputmultiple output (MIMO) digital signal processing (DSP), as previouslydiscussed, but DSP complexity increases with an increasing number ofmodes.

Referring now to FIG. 81, there were illustrated the intensity patternsof the first order mode group within a vortex fiber. Arrows 8102 withinthe illustration show the polarization of the electric field within thefiber. The top row illustrates vector modes that are the exact vectorsolutions, and the bottom row shows the resultant, unstable LP11 modescommonly obtained at a fiber output. Specific linear combinations ofpairs of top row modes resulting in the variety of LP11 modes obtainedat the fiber output. Coupled mode 8102 is provided by the coupled pairof mode 8104 and 8106. Coupled mode 8104 is provided by the coupled pairof mode 8104 and mode 8108. Coupled mode 8116 is provided by the coupledpair of mode 8106 and mode 8110, and coupled mode 8118 is provided bythe coupled pair of mode 8108 and mode 8110.

Typically, index separation of two polarizations and single mode fibersis on the order of 10-7. While this small separation lowers the PMD ofthe fiber, external perturbations can easily couple one mode intoanother, and indeed in a single mode fiber, arbitrary polarizations aretypically observed at the output. Simple fiber polarization controllerthat uses stress induced birefringence can be used to achieve anydesired polarization at the output of the fiber.

By the origin, mode coupling can be classified as distributed (caused byrandom perturbations in fibers), or discrete (caused at the modalcouplers and the multiplexers). Most importantly, it has been shown thatsmall, effective index separation among higher order modes is the mainreason for mode coupling and mode instabilities. In particular, thedistributed mode coupling has been shown to be inversely proportional toΔ-P with P greater than 4, depending on coupling conditions. Modeswithin one group are degenerate. For this reason, in most multimodefiber modes that are observed in the fiber output are in fact the linearcombinations of vector modes and are linearly polarized states. Hence,optical angular momentum modes that are the linear combination of the HEeven, odd modes cannot coexist in these fibers due to coupling todegenerate TE01 and TM01 states.

Thus, the combination of the various OAM modes is not likely to generatemodal coupling within the optical systems and by increasing the numberof OAM modes, the reduction in mode coupling is further benefited.

Referring now to FIGS. 82A and 82B, there is illustrated the benefit ofeffective index separation in first order modes. FIG. 82A illustrates atypical step index multimode fiber that does not exhibit effective indexseparation causing mode coupling. The mode TM₀₁ HE^(even) ₂₁, modeHE^(odd) ₂₁, and mode TE₀₁ have little effective index separation, andthese modes would be coupled together. Mode HE^(x,l) ₁₁ has an effectiveindex separation such that this mode is not coupled with these othermodes.

This can be compared with the same modes in FIG. 82B. In this case,there is an effective separation 8202 between the TM₀₁ mode and theHE^(even) ₂₁ mode and the TE₀₁ mode and the HE^(odd) ₂₁ mode. Thiseffective separation causes no mode coupling between these mode levelsin a similar manner that was done in the same modes in FIG. 82A.

In addition to effective index separation, mode coupling also depends onthe strength of perturbation. An increase in the cladding diameter of anoptical fiber can reduce the bend induced perturbations in the fiber.Special fiber design that includes the trench region can achieveso-called bend insensitivity, which is predominant in fiber to the home.Fiber design that demonstrates reduced bends and sensitivity of higherorder Bessel modes for high power lasers have been demonstrated. Mostimportant, a special fiber design can remove the degeneracy of the firstorder mode, thus reducing the mode coupling and enabling the OAM modesto propagate within these fibers.

Topological charge may be multiplexed to the wave length for eitherlinear or circular polarization. In the case of linear polarizations,topological charge would be multiplexed on vertical and horizontalpolarization. In case of circular polarization, topological charge wouldbe multiplexed on left hand and right hand circular polarization.

The topological charges can be created using Spiral Phase Plates (SPPs)such as that illustrated in FIG. 11e , phase mask holograms or a SpatialLight Modulator (SLM) by adjusting the voltages on SLM which createsproperly varying index of refraction resulting in twisting of the beamwith a specific topological charge. Different topological charges can becreated and muxed together and de-muxed to separate charges. Whensignals are muxed together, multiple signals having different orthogonalfunctions or helicities applied thereto are located in a same signal.The muxed signals are spatially combined in a same signal.

As Spiral Phase plates can transform a plane wave (l=0) to a twistedwave of a specific helicity (i.e. l=+1), Quarter Wave Plates (QWP) cantransform a linear polarization (s=0) to circular polarization (i.e.s=+1).

Cross talk and multipath interference can be reduced usingMultiple-Input-Multiple-Output (MIMO).

Most of the channel impairments can be detected using a control or pilotchannel and be corrected using algorithmic techniques (closed loopcontrol system).

Optical Fiber Communications Using OAM Multiplexing

OAM multiplexing may be implemented in fiber communications. OAM modesare essentially a group of higher order modes defined on a differentbasis as compared to other forms of modes in fiber, such as “linearlypolarized” (LP) modes and fiber vector modes. In principle each of themode sets form an orthogonal mode basis spanning the spatial domain, andmay be used to transmit different data channels. Both LP modes and OAMmodes face challenges of mode coupling when propagating in a fiber, andmay also cause channel crosstalk problems.

In general, two approaches may be involved in fiber transmission usingOAM multiplexing. The first approach is to implement OAM transmission ina regular few mode fiber such as that illustrated in FIG. 78. As is thecase of SDM using LP modes, MIMO DSP is generally required to equalizethe channel interface. The second approach is to utilize a speciallydesigned vortex fiber that suffers from less mode coupling, and DSPequalization can therefore be saved for a certain distance oftransmission.

OAM Transmission in Regular Few Mode Fiber

In a regular few mode fiber, each OAM mode represents approximately alinear combination of the true fiber modes (the solution to the waveequation in fiber). For example, as illustrated in FIG. 83, a linearlypolarized OAM beam 8302 with l=+1 comprises the components of Eigenmodes including TE₀₁, TM₀₁ and HE₂₁. Due to the perturbations or othernon-idealities, OAM modes that are launched into a few mode fiber (FMF)may quickly coupled to each other, most likely manifesting in a group ofLP modes at the fiber output. The mutual mode coupling in fiber may leadto inter-channel crosstalk and eventually failure of the transmission.One possible solution for the mode coupling effects is to use MIMO DSPin combination with coherent detection.

Referring now to FIG. 84, there is illustrated a demonstration of thetransmission of four OAM beams (l=+1 and −1 each with 2 orthogonalpolarization states), each carrying 20 Gbit/s QPSK data, in anapproximately 5 kilometer regular FMF (few mode fiber) 8404. Four datachannels 8402 (2 with x-pol and 2 with y-pol) were converted topol-muxed OAM beams with l=+1 and −1 using an inverse mode sorter 8406.The pol-muxed to OAM beams 8408 (four in total) are coupled into the FMF8404 for propagation. At the fiber output, the received modes weredecomposed onto an OAM basis (l=+1 and −1) using a mode sorter 8410. Ineach of the two OAM components of light were coupled onto a fiber-basedPBS for polarization demultiplexing. Each output 8412 is detected by aphotodiode, followed by ADC (analog-to-digital converter) and off-lineprocessing. To mitigate the inter-channel interference, a constantmodulus algorithm is used to blindly estimate the channel crosstalk andcompensate for the inter-channel interference using linear equalization.Eventually, the QPSK data carried on each OAM beam is recovered with theassistance of a MIMO DSP as illustrated in FIGS. 85A and 85B.

OAM Transmission in a Vortex Fiber

A key challenge for OAM multiplexing in conventional fibers is thatdifferent OAM modes tend to couple to each other during thetransmission. The major reason for this is that in a conventional fiberOAM modes have a relatively small effective refractive index difference(Δ n_(eff)). Stably transmitting an OAM mode in fiber requires somemodifications of the fiber. One manner for stably transmitting OAM modesuses a vortex fiber such as that illustrated in FIG. 86. A vortex fiber8602 is a specially designed a few mode fiber including an additionalhigh index ring 8604 around the fiber core 8606. The design increasesthe effective index differences of modes and therefore reduces themutual mode coupling.

Using this vortex fiber 8602, two OAM modes with l=+1 and −1 and twopolarizations multiplexed fundamental modes were transmitted togetherfor 1.1 km. The measured mode cross talk between two OAM modes wasapproximately −20 dB. These four distinct modes were used to eachcarried a 100 Gbuad QPSK signal at the same wavelength andsimultaneously propagate in the vortex fiber. After the modedemultiplexing, all data was recovered with a power penalty ofapproximately 4.1 dB, which could be attributed to the multipath effectsand mode cross talk. In a further example, WDM was added to furtherextend the capacity of a vortex fiber transmission system. A 20 channelfiber link using to OAM modes and 10 WDM channels (from 1546.642 nm to1553.88 nm), each channel sending 80 Gb/s 16-QAM signal wasdemonstrated, resulting in a total transmission capacity of 1.2 Tb/sunder the FEC limit.

There are additional innovative efforts being made to design andfabricate fibers that are more suitable for OAM multiplexing. A recentlyreported air-core fiber has been demonstrated to further increase therefractive index difference of eigenmodes such that the fiber is able tostably transmit 12 OAM states (l=±7, ±8 and ±9, each with two orthogonalpolarizations) for 2 m. A few mode fibers having an inverse parabolicgraded index profile in which propagating 8 OAM orders (l=±1 and ±2,each with two orthogonal polarizations) has been demonstrated over 1.1km. The same group recently presented a newer version of an air corefiber, whereby the supported OAM states was increased to 16. Onepossible design that can further increase the supported OAM modes and afiber is to use multiple high contrast indexed ring core structure whichis indicated a good potential for OAM multiplexing for fibercommunications.

RF Communications with OAM

As a general property of electromagnetic waves, OAM can also be carriedon other ways with either a shorter wavelength (e.g., x-ray), or alonger wavelength (millimeter waves and terahertz waves) than an opticalbeam. Focusing on the RF waves, OAM beams at 90 GHz were initiallygenerated using a spiral phase plate made of Teflon. Differentapproaches, such as a phase array antenna and a helicoidal parabolicantenna have also been proposed. RF OAM beams have been used as datacarriers for RF communications. A Gaussian beam and an OAM beam withl=+1 at approximately 2.4 GHz have been transmitted by a Yagi-Udaantenna and a spiral parabolic antenna, respectively, which are placedin parallel. These two beams were distinguished by the differentialoutput of a pair of antennas at the receiver side. The number ofchannels was increased to three (carried on OAM beams with l=−1, 0 and+1) using a similar apparatus to send approximately 11 Mb/s signal atapproximately 17 GHz carrier. Note that in these two demonstrationsdifferent OAM beams propagate along different spatial axes. There aresome potential benefits if all of the OAM beams are actually multiplexedand propagated through the same aperture. In a recent demonstrationeight polarization multiplexed (pol-muxed) RF OAM beams (for OAM beamson each of two orthogonal polarizations) our coaxially propagatedthrough a 2.5 m link.

The herein described RF techniques have application in a wide variety ofRF environments. These include RF Point to Point/Multipointapplications, RF Point to Point Backhaul applications, RF Point to PointFronthaul applications (these provide higher throughput CPRI interfacefor cloudification and virtualization of RAN and future cloudifiedHetNet), RF Sattellite applications, RF Wifi (LAN) applications, RFBluetooth (PAN) applications, RF personal device cable replacementapplications, RF Radar applications and RF electromagnet tagapplications. The techniques could also be used in a RF and FSO hybridsystem that can provide communications in an RF mode or an FSO modedepending on which mode of operation is providing the most optimal orcost effective communications link at a particular point in time.

The four different OAM beams with l=−3, −1, +1 and +3 on each of 2orthogonal polarizations are generated using customized spiral phaseplates specifically for millimeter wave at 28 GHz. The observedintensity profile for each of the beams and their interferograms areshown in FIG. 87. These OAM beams were coaxially multiplexed usingdesigned beam splitters. After propagation, the OAM channels weremultiplexed using an inverse spiral phase plate and a spatial filter(the receiver antenna). The measured crosstalk it 28 GHz for each of thedemultiplexed channels is shown in Table 8. It can be seen that thecross talk is low enough for 16-QAM data transmission without theassistance of extra DSPs to reduce the channel interference.

TABLE 8 Crosstalk of the OAM channels measured at f = 28 GHz (CW) l = −3l = −1 f = +1 l = +3 Single-pol (Y-pol) −25 dB   −23 dB   −25 dB −26 dBDual-pol (X-pol) −17 dB −16.5 dB −18.1 dB −19 dB Dual-pol (Y-pol) −18 dB−16.5 dB −16.5 dB −24 dB

Considering that each beam carries a 1 Gbaud 16-QAM signal, a total linkcapacity of 32 Gb/s at a single carrier frequency of 28 GHz and aspectral efficiency of 16 Gb/s/Hz may be achieved. In addition, an RFOAM beam demultiplexer (“mode sorter”) was also customize for a 28 GHzcarrier and is implemented in such a link to simultaneously separatemultiple OAM beams. Simultaneously demultiplexing for OAM beams at thesingle polarization has been demonstrated with a cross talk of less than−14 dB. The cross talk is likely to be further reduced by optimizing thedesign parameters.

Free Space Communications

An additional configuration in which the optical angular momentumprocessing and multi-layer overlay modulation technique described hereinabove may prove useful within the optical network framework is use withfree-space optics communications. Free-space optics systems provide anumber of advantages over traditional UHF RF based systems from improvedisolation between the systems, the size and the cost of thereceivers/transmitters, lack of RF licensing laws, and by combiningspace, lighting, and communication into the same system. Referring nowto FIG. 88 there is illustrated an example of the operation of afree-space communication system. The free-space communication systemutilizes a free-space optics transmitter 8802 that transmits a lightbeam 8804 to a free-space optics receiver 8806. The major differencebetween a fiber-optic network and a free-space optic network is that theinformation beam is transmitted through free space rather than over afiber-optic cable. This causes a number of link difficulties, which willbe more fully discussed herein below. Free-space optics is a line ofsight technology that uses the invisible beams of light to provideoptical bandwidth connections that can send and receive up to 2.5 Gbpsof data, voice, and video communications between a transmitter 8802 anda receiver 8806. Free-space optics uses the same concepts asfiber-optics, except without the use of a fiber-optic cable. Free-spaceoptics systems provide the light beam 8804 within the infrared (IR)spectrum, which is at the low end of the light spectrum. Specifically,the optical signal is in the range of 300 Gigahertz to 1 Terahertz interms of wavelength.

Presently existing free-space optics systems can provide data rates ofup to 10 Gigabits per second at a distance of up to 2.5 kilometers. Inouter space, the communications range of free space opticalcommunications is currently on the order of several thousand kilometers,but has the potential to bridge interplanetary distances of millions ofkilometers, using optical telescopes as beam expanders. In January of2013, NASA used lasers to beam an image of the Mona Lisa to the LunarReconnaissance Orbiter roughly 240,000 miles away. To compensate foratmospheric interference, an error correction code algorithm, similar tothat used within compact discs, was implemented.

The distance records for optical communications involve detection andemission of laser light by space probes. A two-way distance record forcommunication was established by the Mercury Laser Altimeter instrumentaboard the MESSENGER spacecraft. This infrared diode neodymium laser,designed as a laser altimeter for a Mercury Orbiter mission, was able tocommunicate across a distance of roughly 15,000,000 miles (24,000,000kilometers) as the craft neared Earth on a fly by in May of 2005. Theprevious record had been set with a one-way detection of laser lightfrom Earth by the Galileo Probe as two ground based lasers were seenfrom 6,000,000 kilometers by the outbound probe in 1992. Researchersused a white LED based space lighting system for indoor local areanetwork communications.

Referring now to FIG. 89, there is illustrated a block diagram of afree-space optics system using orbital angular momentum and multileveloverlay modulation according to the present disclosure. The OAM twistedsignals, in addition to being transmitted over fiber, may also betransmitted using free optics. In this case, the transmission signalsare generated within transmission circuitry 8902 at each of the FSOtransceivers 8904. Free-space optics technology is based on theconnectivity between the FSO based optical wireless units, eachconsisting of an optical transceiver 8904 with a transmitter 8902 and areceiver 8906 to provide full duplex open pair and bidirectional closedpairing capability. Each optical wireless transceiver unit 8904additionally includes an optical source 8908 plus a lens or telescope8910 for transmitting light through the atmosphere to another lens 8910receiving the information. At this point, the receiving lens ortelescope 8910 connects to a high sensitivity receiver 8906 via opticalfiber 8912. The transmitting transceiver 8904 a and the receivingtransceiver 8904 b have to have line of sight to each other. Trees,buildings, animals, and atmospheric conditions all can hinder the lineof sight needed for this communications medium. Since line of sight isso critical, some systems make use of beam divergence or a diffused beamapproach, which involves a large field of view that toleratessubstantial line of sight interference without significant impact onoverall signal quality. The system may also be equipped with autotracking mechanism 8914 that maintains a tightly focused beam on thereceiving transceiver 3404 b, even when the transceivers are mounted ontall buildings or other structures that sway.

The modulated light source used with optical source 8908 is typically alaser or light emitting diode (LED) providing the transmitted opticalsignal that determines all the transmitter capabilities of the system.Only the detector sensitivity within the receiver 8906 plays an equallyimportant role in total system performance. For telecommunicationspurposes, only lasers that are capable of being modulated at 20 Megabitsper second to 2.5 Gigabits per second can meet current marketplacedemands. Additionally, how the device is modulated and how muchmodulated power is produced are both important to the selection of thedevice. Lasers in the 780-850 nm and 1520-1600 nm spectral bands meetfrequency requirements.

Commercially available FSO systems operate in the near IR wavelengthrange between 750 and 1600 nm, with one or two systems being developedto operate at the IR wavelength of 10,000 nm. The physics andtransmissions properties of optical energy as it travels through theatmosphere are similar throughout the visible and near IR wavelengthrange, but several factors that influence which wavelengths are chosenfor a particular system.

The atmosphere is considered to be highly transparent in the visible andnear IR wavelength. However, certain wavelengths or wavelength bands canexperience severe absorption. In the near IR wavelength, absorptionoccurs primarily in response to water particles (i.e., moisture) whichare an inherent part of the atmosphere, even under clear weatherconditions. There are several transmission windows that are nearlytransparent (i.e., have an attenuation of less than 0.2 dB perkilometer) within the 700-10,000 nm wavelength range. These wavelengthsare located around specific center wavelengths, with the majority offree-space optics systems designed to operate in the windows of 780-850nm and 1520-1600 nm.

Wavelengths in the 780-850 nm range are suitable for free-space opticsoperation and higher power laser sources may operate in this range. At780 nm, inexpensive CD lasers may be used, but the average lifespan ofthese lasers can be an issue. These issues may be addressed by runningthe lasers at a fraction of their maximum rated output power which willgreatly increase their lifespan. At around 850 nm, the optical source8908 may comprise an inexpensive, high performance transmitter anddetector components that are readily available and commonly used innetwork transmission equipment. Highly sensitive silicon (SI) avalanchephotodiodes (APD) detector technology and advanced vertical cavityemitting laser may be utilized within the optical source 8908.

VCSEL technology may be used for operation in the 780 to 850 nm range.Possible disadvantage of this technology include beam detection throughthe use of a night vision scope, although it is still not possible todemodulate a perceived light beam using this technique.

Wavelengths in the 1520-1600 nm range are well-suited for free-spacetransmission, and high quality transmitter and detector components arereadily available for use within the optical source block 8908. Thecombination of low attenuation and high component availability withinthis wavelength range makes the development of wavelength divisionmultiplexing (WDM) free-space optics systems feasible. However,components are generally more expensive and detectors are typically lesssensitive and have a smaller receive surface area when compared withsilicon avalanche photodiode detectors that operator at the 850 nmwavelength. These wavelengths are compatible with erbium-doped fiberamplifier technology, which is important for high power (greater than500 milliwatt) and high data rate (greater than 2.5 Gigabytes persecond) systems. Fifty to 65 times as much power can be transmitted atthe 1520-1600 nm wavelength than can be transmitted at the 780-850 nmwavelength for the same eye safety classification. Disadvantages ofthese wavelengths include the inability to detect a beam with a nightvision scope. The night vision scope is one technique that may be usedfor aligning the beam through the alignment circuitry 8914. Class 1lasers are safe under reasonably foreseeable operating conditionsincluding the use of optical instruments for intrabeam viewing. Class 1systems can be installed at any location without restriction.

Another potential optical source 8908 comprised Class 1M lasers. Class1M laser systems operate in the wavelength range from 302.5 to 4000 nm,which is safe under reasonably foreseeable conditions, but may behazardous if the user employs optical instruments within some portion ofthe beam path. As a result, Class 1M systems should only be installed inlocations where the unsafe use of optical aids can be prevented.Examples of various characteristics of both Class 1 and Class 1M lasersthat may be used for the optical source 4708 are illustrated in Table 9below.

TABLE 9 Laser Power Aperture Size Distance Power Density Classification(mW) (mm) (m) (mW/cm²) 850-nm Wavelength Class 1 0.78 7 14 2.03 50 20000.04 Class 1M 0.78 7 100 2.03 500 7 14 1299.88 50 2000 25.48 1550-nmWavelength Class 1 10 7 14 26.00 25 2000 2.04 Class 1M 10 3.5 100 103.99500 7 14 1299.88 25 2000 101.91

The 10,000 nm wavelength is relatively new to the commercial free spaceoptic arena and is being developed because of better fog transmissioncapabilities. There is presently considerable debate regarding thesecharacteristics because they are heavily dependent upon fog type andduration. Few components are available at the 10,000 nm wavelength, asit is normally not used within telecommunications equipment.Additionally, 10,000 nm energy does not penetrate glass, so it isill-suited to behind window deployment.

Within these wavelength windows, FSO systems should have the followingcharacteristics. The system should have the ability to operate at higherpower levels, which is important for longer distance FSO systemtransmissions. The system should have the ability to provide high speedmodulation, which is important for high speed FSO systems. The systemshould provide a small footprint and low power consumption, which isimportant for overall system design and maintenance. The system shouldhave the ability to operate over a wide temperature range without majorperformance degradations such that the systems may prove useful foroutdoor systems. Additionally, the mean time between failures shouldexceed 10 years. Presently existing FSO systems generally use VCSELS foroperation in the shorter IR wavelength range, and Fabry-Pérot ordistributed feedback lasers for operation in the longer IR wavelengthrange. Several other laser types are suitable for high performance FSOsystems.

A free-space optics system using orbital angular momentum processing andmulti-layer overlay modulation would provide a number of advantages. Thesystem would be very convenient. Free-space optics provides a wirelesssolution to a last-mile connection, or a connection between twobuildings. There is no necessity to dig or bury fiber cable. Free-spaceoptics also requires no RF license. The system is upgradable and itsopen interfaces support equipment from a variety of vendors. The systemcan be deployed behind windows, eliminating the need for costly rooftopright. It is also immune to radiofrequency interference or saturation.The system is also fairly speedy. The system provides 2.5 Gigabits persecond of data throughput. This provides ample bandwidth to transferfiles between two sites. With the growth in the size of files,free-space optics provides the necessary bandwidth to transfer thesefiles efficiently.

Free-space optics also provides a secure wireless solution. The laserbeam cannot be detected with a spectral analyzer or RF meter. The beamis invisible, which makes it difficult to find. The laser beam that isused to transmit and receive the data is very narrow. This means that itis almost impossible to intercept the data being transmitted. One wouldhave to be within the line of sight between the receiver and thetransmitter in order to be able to accomplish this feat. If this occurs,this would alert the receiving site that a connection has been lost.Thus, minimal security upgrades would be required for a free-spaceoptics system.

However, there are several weaknesses with free-space optics systems.The distance of a free-space optics system is very limited. Currentlyoperating distances are approximately within 2 kilometers. Although thisis a powerful system with great throughput, the limitation of distanceis a big deterrent for full-scale implementation. Additionally, allsystems require line of sight be maintained at all times duringtransmission. Any obstacle, be it environmental or animals can hinderthe transmission. Free-space optic technology must be designed to combatchanges in the atmosphere which can affect free-space optic systemperformance capacity.

Something that may affect a free-space optics system is fog. Dense fogis a primary challenge to the operation of free-space optics systems.Rain and snow have little effect on free-space optics technology, butfog is different. Fog is a vapor composed of water droplets which areonly a few hundred microns in diameter, but can modify lightcharacteristics or completely hinder the passage of light through acombination of absorption, scattering, and reflection. The primaryanswer to counter fog when deploying free-space optic based wirelessproducts is through a network design that shortens FSO linked distancesand adds network redundancies.

Absorption is another problem. Absorption occurs when suspended watermolecules in the terrestrial atmosphere extinguish photons. This causesa decrease in the power density (attenuation) of the free space opticsbeam and directly affects the availability of the system. Absorptionoccurs more readily at some wavelengths than others. However, the use ofappropriate power based on atmospheric conditions and the use of spatialdiversity (multiple beams within an FSO based unit), helps maintain therequired level of network availability.

Solar interference is also a problem. Free-space optics systems use ahigh sensitivity receiver in combination with a larger aperture lens. Asa result, natural background light can potentially interfere withfree-space optics signal reception. This is especially the case with thehigh levels of background radiation associated with intense sunlight. Insome instances, direct sunlight may case link outages for periods ofseveral minutes when the sun is within the receiver's field of vision.However, the times when the receiver is most susceptible to the effectsof direct solar illumination can be easily predicted. When directexposure of the equipment cannot be avoided, the narrowing of receiverfield of vision and/or using narrow bandwidth light filters can improvesystem performance. Interference caused by sunlight reflecting off of aglass surface is also possible.

Scattering issues may also affect connection availability. Scattering iscaused when the wavelength collides with the scatterer. The physicalsize of the scatterer determines the type of scattering. When thescatterer is smaller than the wavelength, this is known as Rayleighscattering. When a scatterer is of comparable size to the wavelengths,this is known as Mie scattering. When the scattering is much larger thanthe wavelength, this is known as non-selective scattering. Inscattering, unlike absorption, there is no loss of energy, only adirectional redistribution of energy that may have significant reductionin beam intensity over longer distances.

Physical obstructions such as flying birds or construction cranes canalso temporarily block a single beam free space optics system, but thistends to cause only short interruptions. Transmissions are easily andautomatically resumed when the obstacle moves. Optical wireless productsuse multibeams (spatial diversity) to address temporary abstractions aswell as other atmospheric conditions, to provide for greateravailability.

The movement of buildings can upset receiver and transmitter alignment.Free-space optics based optical wireless offerings use divergent beamsto maintain connectivity. When combined with tracking mechanisms,multiple beam FSO based systems provide even greater performance andenhanced installation simplicity.

Scintillation is caused by heated air rising from the Earth or man-madedevices such as heating ducts that create temperature variations amongdifferent pockets of air. This can cause fluctuations in signalamplitude, which leads to “image dancing” at the free-space optics basedreceiver end. The effects of this scintillation are called “refractiveturbulence.” This causes primarily two effects on the optical beams.Beam wander is caused by the turbulent eddies that are no larger thanthe beam. Beam spreading is the spread of an optical beam as itpropagates through the atmosphere.

Referring now to FIGS. 90A-90D, in order to achieve higher data capacitywithin optical links, an additional degree of freedom from multiplexingmultiple data channels must be exploited. Moreover, the ability to usetwo different orthogonal multiplexing techniques together has thepotential to dramatically enhance system performance and increasedbandwidth.

One multiplexing technique which may exploit the possibilities is modedivision multiplexing (MDM) using orbital angular momentum (OAM). OAMmode refers to laser beams within a free-space optical system orfiber-optic system that have a phase term of e^(ilφ) in their wavefronts, in which φ is the azimuth angle and l determines the OAM value(topological charge). In general, OAM modes have a “donut-like” ringshaped intensity distribution. Multiple spatial collocated laser beams,which carry different OAM values, are orthogonal to each other and canbe used to transmit multiple independent data channels on the samewavelength. Consequently, the system capacity and spectral efficiency interms of bits/S/Hz can be dramatically increased. Free-spacecommunications links using OAM may support 100 Tbits/capacity. Varioustechniques for implementing this as illustrated in FIGS. 90A-90D includea combination of multiple beams 9002 having multiple different OAMvalues 9004 on each wavelength. Thus, beam 9002 includes OAM values,OAM1 and OAM4. Beam 9006 includes OAM value 2 and OAM value 5. Finally,beam 9008 includes OAM3 value and OAM6 value. Referring now to FIG. 90B,there is illustrated a single beam wavelength 9010 using a first groupof OAM values 9012 having both a positive OAM value 9012 and a negativeOAM value 9014. Similarly, OAM2 value may have a positive value 9016 anda negative value 9018 on the same wavelength 9010. While mode divisionmultiplexing of OAM modes is described above, other orthogonal functionsmay be used with mode division multiplexing such as Laguerre Gaussianfunctions, Hermite Gaussian functions, Jacobi functions, Gegenbauerfunctions, Legendre functions and Chebyshev functions.

FIG. 90C illustrates the use of a wavelength 9020 having polarizationmultiplexing of OAM value. The wavelength 9020 can have multiple OAMvalues 9022 multiplexed thereon. The number of available channels can befurther increased by applying left or right handed polarization to theOAM values. Finally, FIG. 90D illustrates two groups of concentric rings9060, 9062 for a wavelength having multiple OAM values.

Wavelength distribution multiplexing (WDM) has been widely used toimprove the optical communication capacity within both fiber-opticsystems and free-space communication system. OAM mode/mode divisionmultiplexing and WDM are mutually orthogonal such that they can becombined to achieve a dramatic increase in system capacity. Referringnow to FIG. 91, there is illustrated a scenario where each WDM channel9102 contains many orthogonal OAM beam 9104. Thus, using a combinationof orbital angular momentum with wave division multiplexing, asignificant enhancement in communication link to capacity may beachieved. By further combining polarization multiplexing with acombination of MDM and WDM even further increased in bandwidth capacitymay be achieved from the +/−polarization values being added to the modeand wavelength multiplexing.

Current optical communication architectures have considerable routingchallenges. A routing protocol for use with free-space optic system musttake into account the line of sight requirements for opticalcommunications within a free-space optics system. Thus, a free-spaceoptics network must be modeled as a directed hierarchical random sectorgeometric graph in which sensors route their data via multi-hop paths toa base station through a cluster head. This is a new efficient routingalgorithm for local neighborhood discovery and a base station uplink anddownlink discovery algorithm. The routing protocol requires orderOlog(n) storage at each node versus order O(n) used within currenttechniques and architectures.

Current routing protocols are based on link state, distance vectors,path vectors, or source routing, and they differ from the new routingtechnique in significant manners. First, current techniques assume thata fraction of the links are bidirectional. This is not true within afree-space optic network in which all links are unidirectional. Second,many current protocols are designed for ad hoc networks in which therouting protocol is designed to support multi-hop communications betweenany pair of nodes. The goal of the sensor network is to route sensorreadings to the base station. Therefore, the dominant traffic patternsare different from those in an ad hoc network. In a sensor network, nodeto base stations, base station to nodes, and local neighborhoodcommunication are mostly used.

Recent studies have considered the effect of unidirectional links andreport that as many as 5 percent to 10 percent of links and wireless adhoc networks are unidirectional due to various factors. Routingprotocols such as DSDV and AODV use a reverse path technique, implicitlyignoring such unidirectional links and are therefore not relevant inthis scenario. Other protocols such as DSR, ZRP, or ZRL have beendesigned or modified to accommodate unidirectionality by detectingunidirectional links and then providing bidirectional abstraction forsuch links. Referring now to FIG. 92, the simplest and most efficientsolution for dealing with unidirectionality is tunneling, in whichbidirectionality is emulated for a unidirectional link by usingbidirectional links on a reverse back channel to establish the tunnel.Tunneling also prevents implosion of acknowledgement packets and loopingby simply pressing link layer acknowledgements for tunneled packetsreceived on a unidirectional link. Tunneling, however, works well inmostly bidirectional networks with few unidirectional links.

Within a network using only unidirectional links such as a free-spaceoptical network, systems such as that illustrated in FIGS. 92 and 93would be more applicable. Nodes within a unidirectional network utilizea directional transmit 9202 transmitting from the node 9200 in a single,defined direction. Additionally, each node 9200 includes anomnidirectional receiver 9204 which can receive a signal coming to thenode in any direction. Also, as discussed here and above, the node 9200would also include a Olog(n) storage 9206. Thus, each node 9200 provideonly unidirectional communications links. Thus, a series of nodes 9200as illustrated in FIG. 93 may unidirectionally communicate with anyother node 9200 and forward communication from one desk location toanother through a sequence of interconnected nodes.

Topological charge may be multiplexed to the wave length for eitherlinear or circular polarization. In the case of linear polarizations,topological charge would be multiplexed on vertical and horizontalpolarization. In case of circular polarization, topological charge wouldbe multiplexed on left hand and right hand circular polarizations.

The topological charges can be created using Spiral Phase Plates (SPPs)such as that illustrated in FIG. 12E, phase mask holograms or a SpatialLight Modulator (SLM) by adjusting the voltages on SLM which createsproperly varying index of refraction resulting in twisting of the beamwith a specific topological charge. Different topological charges can becreated and muxed together and de-muxed to separate charges.

As Spiral Phase plates can transform a plane wave (l=0) to a twistedwave of a specific helicity (i.e. l=+1), Quarter Wave Plates (QWP) cantransform a linear polarization (s=0) to circular polarization (i.e.s=+1).

Cross talk and multipath interference can be reduced usingMultiple-Input-Multiple-Output (MIMO).

Most of the channel impairments can be detected using a control or pilotchannel and be corrected using algorithmic techniques (closed loopcontrol system).

Multiplexing of the topological charge to the RF as well as free spaceoptics in real time provides redundancy and better capacity. Whenchannel impairments from atmospheric disturbances or scintillationimpact the information signals, it is possible to toggle between freespace optics to RF and back in real time. This approach still usestwisted waves on both the free space optics as well as the RF signal.Most of the channel impairments can be detected using a control or pilotchannel and be corrected using algorithmic techniques (closed loopcontrol system) or by toggling between the RF and free space optics.

In a further embodiment illustrated in FIG. 94, both RF signals and freespace optics may be implemented within a dual RF and free space opticsmechanism 9402. The dual RF and free space optics mechanism 9402 includea free space optics projection portion 9404 that transmits a light wavehaving an orbital angular momentum applied thereto with multileveloverlay modulation and a RF portion 9406 including circuitry necessaryfor transmitting information with orbital angular momentum andmultilayer overlay on an RF signal 9410. The dual RF and free spaceoptics mechanism 9402 may be multiplexed in real time between the freespace optics signal 9408 and the RF signal 9410 depending upon operatingconditions. In some situations, the free space optics signal 9408 wouldbe most appropriate for transmitting the data. In other situations, thefree space optics signal 9408 would not be available and the RF signal9410 would be most appropriate for transmitting data. The dual RF andfree space optics mechanism 9402 may multiplex in real time betweenthese two signals based upon the available operating conditions.

Multiplexing of the topological charge to the RF as well as free spaceoptics in real time provides redundancy and better capacity. Whenchannel impairments from atmospheric disturbances or scintillationimpact the information signals, it is possible to toggle between freespace optics to RF and back in real time. This approach still usestwisted waves on both the free space optics as well as the RF signal.Most of the channel impairments can be detected using a control or pilotchannel and be corrected using algorithmic techniques (closed loopcontrol system) or by toggling between the RF and free space optics.

Quantum Communication Using OAM

OAM has also received increasing interest for its potential role in thedevelopment of secure quantum communications that are based on thefundamental laws of quantum mechanics (i.e., quantum no cloningtheorem). One of the examples is high dimensional quantum keydistribution (QKD) QKD systems have conventionally utilized thepolarization or phase of light for encoding. The original proposal forQKD (i.e., the BB 84 protocol of Bennett and Brassard) encodesinformation on the polarization states and so only allow one bit ofinformation to be impressed onto each photon. The benefit of using OAMis that OAM states reside in an infinite dimensional Hilbert space,implying the possibility of encoding multiple bits of information on anindividual photon. Similar to the use of OAM multiplexing in classicaloptical communications, the secure key rate can be further increasedsimultaneous encoding of information in different domains is implementedthrough making use of high dimensional entanglement. The addition to theadvantages of a large alphabet for information encoding, the security ofkeys generated by an OAM-based QKD system have been shown to be improveddue to the use of a large Hilbert space, which indicates increaserobustness of the QKD system against eavesdropping.

FIG. 95 illustrates a seven dimensional QKD link based on OAM encoding.FIG. 96 shows the two complementary seven dimensional bases used forinformation encoding. Recent QKD systems have been demonstrated tooperate at a secure key rate of up to 1 Mb/s. However, in order tosupport an OAM-based QKD system with a higher secure key rate, thedevelopment of a OAM generation methods with speeds higher than MHzwould be required. Another challenge arises from the efficiency inchsorting single photons in the OAM basis, although the current OAMsorting approach allows an OAM separation efficiency of greater than92%. Additionally, adverse channel conditions pose a critical challenge.For a free space QKD system employing OAM states, atmospheric turbulencethat distorts the phase front of an OAM state may significantly degradethe information content of the transmitted OAM light field.

Quantum Key Distribution

Referring now to FIG. 97, there is illustrated a further improvement ofa system utilizing orbital angular momentum processing, LaguerreGaussian processing, Hermite Gaussian processing or processing using anyorthogonal functions. In the illustration of FIG. 97, a transmitter 9702and receiver 9704 are interconnected over an optical link 9706. Theoptical link 9706 may comprise a fiber-optic link or a free-space opticlink as described herein above. The transmitter receives a data stream9708 that is processed via orbital angular momentum processing circuitry9710. The orbital angular momentum processing circuitry 9710 provideorbital angular momentum twist to various signals on separate channelsas described herein above. In some embodiments, the orbital angularmomentum processing circuitry may further provide multi-layer overlaymodulation to the signal channels in order to further increase systembandwidth.

The OAM processed signals are provided to quantum key distributionprocessing circuitry 9712. The quantum key distribution processingcircuitry 9712 utilizes the principals of quantum key distribution aswill be more fully described herein below to enable encryption of thesignal being transmitted over the optical link 9706 to the receiver9704. The received signals are processed within the receiver 9704 usingthe quantum key distribution processing circuitry 9714. The quantum keydistribution processing circuitry 9714 decrypts the received signalsusing the quantum key distribution processing as will be more fullydescribed herein below. The decrypted signals are provided to orbitalangular momentum processing circuitry 9716 which removes any orbitalangular momentum twist from the signals to generate the plurality ofoutput signals 9718. As mentioned previously, the orbital angularmomentum processing circuitry 9716 may also demodulate the signals usingmultilayer overlay modulation included within the received signals.

Orbital angular momentum in combination with optical polarization isexploited within the circuit of FIG. 97 in order to encode informationin rotation invariant photonic states, so as to guarantee fullindependence of the communication from the local reference frames of thetransmitting unit 9702 and the receiving unit 9704. There are variousways to implement quantum key distribution (QKD), a protocol thatexploits the features of quantum mechanics to guarantee unconditionalsecurity in cryptographic communications with error rate performancesthat are fully compatible with real world application environments.

Encrypted communication requires the exchange of keys in a protectedmanner. This key exchanged is often done through a trusted authority.Quantum key distribution is an alternative solution to the keyestablishment problem. In contrast to, for example, public keycryptography, quantum key distribution has been proven to beunconditionally secure, i.e., secure against any attack, even in thefuture, irrespective of the computing power or in any other resourcesthat may be used. Quantum key distribution security relies on the lawsof quantum mechanics, and more specifically on the fact that it isimpossible to gain information about non-orthogonal quantum stateswithout perturbing these states. This property can be used to establishrandom keys between a transmitter and receiver, and guarantee that thekey is perfectly secret from any third party eavesdropping on the line.

In parallel to the “full quantum proofs” mentioned above, the securityof QKD systems has been put on stable information theoretic footing,thanks to the work on secret key agreements done in the framework ofinformation theoretic cryptography and to its extensions, triggered bythe new possibilities offered by quantum information. Referring now toFIG. 98, within a basic QKD system, a QKD link 9802 is a point to pointconnection between a transmitter 9804 and a receiver 9806 that want toshare secret keys. The QKD link 9802 is constituted by the combinationof a quantum channel 9808 and a classic channel 9810. The transmitter9804 generates a random stream of classical bits and encodes them into asequence of non-orthogonal states of light that are transmitted over thequantum channel 9808. Upon reception of these quantum states, thereceiver 9806 performs some appropriate measurements leading thereceiver to share some classical data over the classical link 9810correlated with the transmitter bit stream. The classical channel 9810is used to test these correlations.

If the correlations are high enough, this statistically implies that nosignificant eavesdropping has occurred on the quantum channel 9808 andthus, that has a very high probability, a perfectly secure, symmetrickey can be distilled from the correlated data shared by the transmitter9804 and the receiver 9806. In the opposite case, the key generationprocess has to be aborted and started again. The quantum keydistribution is a symmetric key distribution technique. Quantum keydistribution requires, for authentication purposes, that the transmitter9804 and receiver 9806 share in advance a short key whose length scalesonly logarithmically in the length of the secret key generated by an OKDsession.

Quantum key distribution on a regional scale has already beendemonstrated in a number of countries. However, free-space optical linksare required for long distance communication among areas which are notsuitable for fiber installation or for moving terminals, including theimportant case of satellite based links. The present approach exploitsspatial transverse modes of the optical beam, in particular of the OAMdegree of freedom, in order to acquire a significant technical advantagethat is the insensitivity of the communication to relevant alignment ofthe user's reference frames. This advantage may be very relevant forquantum key distribution implementation to be upgraded from the regionalscale to a national or continental one, or for links crossing hostileground, and even for envisioning a quantum key distribution on a globalscale by exploiting orbiting terminals on a network of satellites.

The OAM Eigen modes are characterized by a twisted wavefront composed of“l” intertwined helices, where “l” is an integer, and by photonscarrying “±lh” of (orbital) angular momentum, in addition to the moreusual spin angular momentum (SAM) associated with polarization. Thepotentially unlimited value of “l” opens the possibility to exploit OAMalso for increasing the capacity of communication systems (although atthe expense of increasing also the channel cross-section size), andterabit classical data transmission based on OAM multiplexing can bedemonstrated both in free-space and optical fibers. Such a feature canalso be exploited in the quantum domain, for example to expand thenumber of qubits per photon, or to achieve new functions, such as therotational invariance of the qubits.

In a free-space QKD, two users (Alice and Bob) must establish a sharedreference frame (SRF) in order to communicate with good fidelity. Indeedthe lack of a SRF is equivalent to an unknown relative rotation whichintroduces noise into the quantum channel, disrupting the communication.When the information is encoded in photon polarization, such a referenceframe can be defined by the orientations of Alice's and Bob's“horizontal” linear polarization directions. The alignment of thesedirections needs extra resources and can impose serious obstacles inlong distance free space QKD and/or when the misalignment varies intime. As indicated, we can solve this by using rotation invariantstates, which remove altogether the need for establishing a SRF. Suchstates are obtained as a particular combination of OAM and polarizationmodes (hybrid states), for which the transformation induced by themisalignment on polarization is exactly balanced by the effect of thesame misalignment on spatial modes. These states exhibit a globalsymmetry under rotations of the beam around its axis and can bevisualized as space-variant polarization states, generalizing thewell-known azimuthal and radial vector beams, and forming atwo-dimensional Hilbert space. Moreover, this rotation-invariant hybridspace can be also regarded as a decoherence-free subspace of thefour-dimensional OAM-polarization product Hilbert space, insensitive tothe noise associated with random rotations.

The hybrid states can be generated by a particular space-variantbirefringent plate having topological charge “q” at its center, named“q-plate”. In particular, a polarized Gaussian beam (having zero OAM)passing through a q-plate with q=½ will undergo the followingtransformation:

(α|R

+β|

)_(π)

|0

₀→α|L

_(π)

|

+β|R)π

|L

₀

|L>_(π) _(_) and |R>_(π) denote the left and right circular polarizationstates (eigenstates of SAM with eigenvalues “±

”), |0>_(O) represents the transverse Gaussian mode with zero OAM andthe |L>_(O) _(_) and |R>_(O) eigenstates of OAM with |l|=1 and witheigenvalues “±l

”). The states appearing on the right hand side of equation arerotation-invariant states. The reverse operation to this can be realizedby a second q-plate with the same q. In practice, the q-plate operatesas an interface between the polarization space and the hybrid one,converting qubits from one space to the other and vice versa in auniversal (qubit invariant) way. This in turn means that the initialencoding and final decoding of information in our QKD implementationprotocol can be conveniently performed in the polarization space, whilethe transmission is done in the rotation-invariant hybrid space.

OAM is a conserved quantity for light propagation in vacuum, which isobviously important for communication applications. However, OAM is alsohighly sensitive to atmospheric turbulence, a feature which limits itspotential usefulness in many practical cases unless new techniques aredeveloped to deal with such issues.

Quantum cryptography describes the use of quantum mechanical effects (inparticular quantum communication and quantum computation) to performcryptographic tasks or to break cryptographic systems. Well-knownexamples of quantum cryptography are the use of quantum communication toexchange a key securely (quantum key distribution) and the hypotheticaluse of quantum computers that would allow the breaking of variouspopular public-key encryption and signature schemes (e.g., RSA).

The advantage of quantum cryptography lies in the fact that it allowsthe completion of various cryptographic tasks that are proven to beimpossible using only classical (i.e. non-quantum) communication. Forexample, quantum mechanics guarantees that measuring quantum datadisturbs that data; this can be used to detect eavesdropping in quantumkey distribution.

Quantum key distribution (QKD) uses quantum mechanics to guaranteesecure communication. It enables two parties to produce a shared randomsecret key known only to them, which can then be used to encrypt anddecrypt messages.

An important and unique property of quantum distribution is the abilityof the two communicating users to detect the presence of any third partytrying to gain knowledge of the key. This results from a fundamentalaspect of quantum mechanics: the process of measuring a quantum systemin general disturbs the system. A third party trying to eavesdrop on thekey must in some way measure it, thus introducing detectable anomalies.By using quantum superposition or quantum entanglement and transmittinginformation in quantum states, a communication system can be implementedwhich detects eavesdropping. If the level of eavesdropping is below acertain threshold, a key can be produced that is guaranteed to be secure(i.e. the eavesdropper has no information about it), otherwise no securekey is possible and communication is aborted.

The security of quantum key distribution relies on the foundations ofquantum mechanics, in contrast to traditional key distribution protocolwhich relies on the computational difficulty of certain mathematicalfunctions, and cannot provide any indication of eavesdropping orguarantee of key security.

Quantum key distribution is only used to reduce and distribute a key,not to transmit any message data. This key can then be used with anychosen encryption algorithm to encrypt (and decrypt) a message, which istransmitted over a standard communications channel. The algorithm mostcommonly associated with QKD is the one-time pad, as it is provablysecure when used with a secret, random key.

Quantum communication involves encoding information in quantum states,or qubits, as opposed to classical communication's use of bits. Usually,photons are used for these quantum states and thus is applicable withinoptical communication systems. Quantum key distribution exploits certainproperties of these quantum states to ensure its security. There areseveral approaches to quantum key distribution, but they can be dividedinto two main categories, depending on which property they exploit. Thefirst of these are prepare and measure protocol. In contrast toclassical physics, the act of measurement is an integral part of quantummechanics. In general, measuring an unknown quantum state changes thatstate in some way. This is known as quantum indeterminacy, and underliesresults such as the Heisenberg uncertainty principle, informationdistribution theorem, and no cloning theorem. This can be exploited inorder to detect any eavesdropping on communication (which necessarilyinvolves measurement) and, more importantly, to calculate the amount ofinformation that has been intercepted. Thus, by detecting the changewithin the signal, the amount of eavesdropping or information that hasbeen intercepted may be determined by the receiving party.

The second category involves the use of entanglement based protocols.The quantum states of two or more separate objects can become linkedtogether in such a way that they must be described by a combined quantumstate, not as individual objects. This is known as entanglement, andmeans that, for example, performing a measurement on one object affectsthe other object. If an entanglement pair of objects is shared betweentwo parties, anyone intercepting either object alters the overallsystem, revealing the presence of a third party (and the amount ofinformation that they have gained). Thus, again, undesired reception ofinformation may be determined by change in the entangled pair of objectsthat is shared between the parties when intercepted by an unauthorizedthird party.

One example of a quantum key distribution (QKD) protocol is the BB84protocol. The BB84 protocol was originally described using photonpolarization states to transmit information. However, any two pairs ofconjugate states can be used for the protocol, and optical fiber-basedimplementations described as BB84 can use phase-encoded states. Thetransmitter (traditionally referred to as Alice) and the receiver(traditionally referred to as Bob) are connected by a quantumcommunication channel which allows quantum states to be transmitted. Inthe case of photons, this channel is generally either an optical fiber,or simply free-space, as described previously with respect to FIG. 97.In addition, the transmitter and receiver communicate via a publicclassical channel, for example using broadcast radio or the Internet.Neither of these channels needs to be secure. The protocol is designedwith the assumption that an eavesdropper (referred to as Eve) caninterfere in any way with both the transmitter and receiver.

Referring now to FIG. 99, the security of the protocol comes fromencoding the information in non-orthogonal states. Quantum indeterminacymeans that these states cannot generally be measured without disturbingthe original state. BB84 uses two pair of states 9902, each pairconjugate to the other pair to form a conjugate pair 9904. The twostates 9902 within a pair 9904 are orthogonal to each other. Pairs oforthogonal states are referred to as a basis. The usual polarizationstate pairs used are either the rectilinear basis of vertical (0degrees) and horizontal (90 degrees), the diagonal basis of 45 degreesand 135 degrees, or the circular basis of left handedness and/or righthandedness. Any two of these basis are conjugate to each other, and soany two can be used in the protocol. In the example of FIG. 100,rectilinear basis are used at 10002 and 10004, respectively, anddiagonal basis are used at 10006 and 10008.

The first step in BB84 protocol is quantum transmission. Referring nowto FIG. 101 wherein there is illustrated a flow diagram describing theprocess, wherein the transmitter creates a random bit (0 or 1) at step10102, and randomly selects at 10104 one of the two basis, eitherrectilinear or diagonal, to transmit the random bit. The transmitterprepares at step 10106 a photon polarization state depending both on thebit value and the selected basis, as shown in FIG. 55. So, for example,a 0 is encoded in the rectilinear basis (+) as a vertical polarizationstate and a 1 is encoded in a diagonal basis (X) as a 135 degree state.The transmitter transmits at step 10108 a single proton in the statespecified to the receiver using the quantum channel. This process isrepeated from the random bit stage at step 10102 with the transmitterrecording the state, basis, and time of each photon that is sent overthe optical link.

According to quantum mechanics, no possible measurement distinguishesbetween the four different polarization states 10002 through 10008 ofFIG. 100, as they are not all orthogonal. The only possible measurementis between any two orthogonal states (and orthonormal basis). So, forexample, measuring in the rectilinear basis gives a result of horizontalor vertical. If the photo was created as horizontal or vertical (as arectilinear eigenstate), then this measures the correct state, but if itwas created as 45 degrees or 135 degrees (diagonal eigenstate), therectilinear measurement instead returns either horizontal or vertical atrandom. Furthermore, after this measurement, the proton is polarized inthe state it was measured in (horizontal or vertical), with all of theinformation about its initial polarization lost.

Referring now to FIG. 102, as the receiver does not know the basis thephotons were encoded in, the receiver can only select a basis at randomto measure in, either rectilinear or diagonal. At step 10202, thetransmitter does this for each received photon, recording the timemeasurement basis used and measurement result at step 10204. At step10206, a determination is made if there are further protons present and,if so, control passes back to step 10202. Once inquiry step 10206determines the receiver had measured all of the protons, the transceivercommunicates at step 10208 with the transmitter over the publiccommunications channel. The transmitter broadcast the basis for eachphoton that was sent at step 10210 and the receiver broadcasts the basiseach photon was measured in at step 10212. Each of the transmitter andreceiver discard photon measurements where the receiver used a differentbasis at step 10214 which, on average, is one-half, leaving half of thebits as a shared key, at step 10216. This process is more fullyillustrated in FIG. 103.

The transmitter transmits the random bit 01101001. For each of thesebits respectively, the transmitter selects the sending basis ofrectilinear, rectilinear, diagonal, rectilinear, diagonal, diagonal,diagonal, and rectilinear. Thus, based upon the associated random bitsselected and the random sending basis associated with the signal, thepolarization indicated in line 10202 is provided. Upon receiving thephoton, the receiver selects the random measuring basis as indicated inline 10304. The photon polarization measurements from these basis willthen be as indicated in line 10306. A public discussion of thetransmitted basis and the measurement basis are discussed at 10308 andthe secret key is determined to be 0101 at 10310 based upon the matchingbases for transmitted photons 1, 3, 6, and 8.

Referring now to FIG. 104, there is illustrated the process fordetermining whether to keep or abort the determined key based uponerrors detected within the determined bit string. To check for thepresence of eavesdropping, the transmitter and receiver compare acertain subset of their remaining bit strings at step 10402. If a thirdparty has gained any information about the photon's polarization, thisintroduces errors within the receiver's measurements. If more than Pbits differ at inquiry step 10404, the key is aborted at step 10406, andthe transmitter and receiver try again, possibly with a differentquantum channel, as the security of the key cannot be guaranteed. P ischosen so that if the number of bits that is known to the eavesdropperis less than this, privacy amplification can be used to reduce theeavesdropper's knowledge of the key to an arbitrarily small amount byreducing the length of the key. If inquiry step 10404 determines thatthe number of bits is not greater than P, then the key may be used atstep 10408.

The E91 protocol comprises another quantum key distribution scheme thatuses entangled pairs of protons. This protocol may also be used withentangled pairs of protons using orbital angular momentum processing,Laguerre Gaussian processing, Hermite Gaussian processing or processingusing any orthogonal functions for Q-bits. The entangled pairs can becreated by the transmitter, by the receiver, or by some other sourceseparate from both of the transmitter and receiver, including aneavesdropper. The photons are distributed so that the transmitter andreceiver each end up with one photon from each pair. The scheme relieson two properties of entanglement. First, the entangled states areperfectly correlated in the sense that if the transmitter and receiverboth measure whether their particles have vertical or horizontalpolarizations, they always get the same answer with 100 percentprobability. The same is true if they both measure any other pair ofcomplementary (orthogonal) polarizations. However, the particularresults are not completely random. It is impossible for the transmitterto predict if the transmitter, and thus the receiver, will get verticalpolarizations or horizontal polarizations. Second, any attempt ateavesdropping by a third party destroys these correlations in a way thatthe transmitter and receiver can detect. The original Ekert protocol(E91) consists of three possible states and testing Bell inequalityviolation for detecting eavesdropping.

Presently, the highest bit rate systems currently using quantum keydistribution demonstrate the exchange of secure keys at 1 Megabit persecond over a 20 kilometer optical fiber and 10 Kilobits per second overa 100 kilometer fiber.

The longest distance over which quantum key distribution has beendemonstrated using optical fiber is 148 kilometers. The distance is longenough for almost all of the spans found in today's fiber-opticnetworks. The distance record for free-space quantum key distribution is144 kilometers using BB84 enhanced with decoy states.

Referring now to FIG. 105, there is illustrated a functional blockdiagram of a transmitter 10502 and receiver 10504 that can implementalignment of free-space quantum key distribution. The system canimplement the BB84 protocol with decoy states. The controller 10506enables the bits to be encoded in two mutually unbiased bases Z={|0>,|1>} and X={|+>, |−>}, where |0> and |1> are two orthogonal statesspanning the qubit space and |±

=1/√2 (|0)±|1

). The transmitter controller 10506 randomly chooses between the Z and Xbasis to send the classical bits 0 and 1. Within hybrid encoding, the Zbasis corresponds to {|L)_(π)

|r

_(O), |R

_(π)

|l

_(O)} while the X basis states correspond to 1/√2 (|L)_(π)

|r)_(O)±|R

_(π)

|l

_(O)). The transmitter 10502 uses four different polarized attenuatedlasers 10508 to generate quantum bits through the quantum bit generator10510. Photons from the quantum bit generator 41050 are delivered via asingle mode fiber 10512 to a telescope 10514. Polarization states |H>,|V>, |R>, |L> are transformed into rotation invariant hybrid states bymeans of a q-plate 10516 with q=½. The photons can then be transmittedto the receiving station 10504 where a second q-plate transform 10518transforms the signals back into the original polarization states |H>,|V>, |R>, |L>, as defined by the receiver reference frame. Qubits canthen be analyzed by polarizers 10520 and single photon detectors 10522.The information from the polarizers 10520 and photo detectors 10522 maythen be provided to the receiver controller 10524 such that the shiftedkeys can be obtained by keeping only the bits corresponding to the samebasis on the transmitter and receiver side as determined bycommunications over a classic channel between the transceivers 10526,10528 in the transmitter 10502 and receiver 10504.

Referring now to FIG. 106, there is illustrated a network cloud basedquantum key distribution system including a central server 10602 andvarious attached nodes 10604 in a hub and spoke configuration. Trends innetworking are presenting new security concerns that are challenging tomeet with conventional cryptography, owing to constrained computationalresources or the difficulty of providing suitable key management. Inprinciple, quantum cryptography, with its forward security andlightweight computational footprint, could meet these challenges,provided it could evolve from the current point to point architecture toa form compatible with multimode network architecture. Trusted quantumkey distribution networks based on a mesh of point to point links lacksscalability, require dedicated optical fibers, are expensive and notamenable to mass production since they only provide one of thecryptographic functions, namely key distribution needed for securecommunications. Thus, they have limited practical interest.

A new, scalable approach such as that illustrated in FIG. 106 providesquantum information assurance that is network based quantumcommunications which can solve new network security challenges. In thisapproach, a BB84 type quantum communication between each of N clientnodes 10604 and a central sever 10602 at the physical layer support aquantum key management layer, which in turn enables secure communicationfunctions (confidentiality, authentication, and nonrepudiation) at theapplication layer between approximately N2 client pairs. This networkbased communication “hub and spoke” topology can be implemented in anetwork setting, and permits a hierarchical trust architecture thatallows the server 10602 to act as a trusted authority in cryptographicprotocols for quantum authenticated key establishment. This avoids thepoor scaling of previous approaches that required a pre-existing trustrelationship between every pair of nodes. By making a server 10602, asingle multiplex QC (quantum communications) receiver and the clientnodes 10604 QC transmitters, this network can simplify complexity acrossmultiple network nodes. In this way, the network based quantum keydistribution architecture is scalable in terms of both quantum physicalresources and trust. One can at time multiplex the server 10602 withthree transmitters 10604 over a single mode fiber, larger number ofclients could be accommodated with a combination of temporal andwavelength multiplexing as well as orbital angular momentum multiplexedwith wave division multiplexing to support much higher clients.

Referring now to FIGS. 107 and 108, there are illustrated variouscomponents of multi-user orbital angular momentum based quantum keydistribution multi-access network. FIG. 107 illustrates a high speedsingle photon detector 10702 positioned at a network node that can beshared between multiple users 10704 using conventional networkarchitectures, thereby significantly reducing the hardware requirementsfor each user added to the network. In an embodiment, the single photondetector 10702 may share up to 64 users. This shared receiverarchitecture removes one of the main obstacles restricting thewidespread application of quantum key distribution. The embodimentpresents a viable method for realizing multi-user quantum keydistribution networks with resource efficiency.

Referring now also to FIG. 108, in a nodal quantum key distributionnetwork, multiple trusted repeaters 10802 are connected via point topoint links 10804 between node 10806. The repeaters are connected viapoint to point links between a quantum transmitter and a quantumreceiver. These point to point links 10804 can be realized using longdistance optical fiber lengths and may even utilize ground to satellitequantum key distribution communication. While point to point connections10804 are suitable to form a backbone quantum core network, they areless suitable to provide the last-mile service needed to give amultitude of users access to the quantum key distributioninfrastructure. Reconfigurable optical networks based on opticalswitches or wavelength division multiplexing may achieve more flexiblenetwork structures, however, they also require the installation of afull quantum key distribution system per user which is prohibitivelyexpensive for many applications.

The quantum key signals used in quantum key distribution need onlytravel in one direction along a fiber to establish a secure key betweenthe transmitter and the receiver. Single photon quantum key distributionwith the sender positioned at the network node 10806 and the receiver atthe user premises therefore lends itself to a passive multi-user networkapproach. However, this downstream implementation has two majorshortcomings. Firstly, every user in the network requires a singlephoton detector, which is often expensive and difficult to operate.Additionally, it is not possible to deterministically address a user.All detectors, therefore, have to operate at the same speed as atransmitter in order not to miss photons, which means that most of thedetector bandwidth is unused.

Most systems associated with a downstream implementation can beovercome. The most valuable resource should be shared by all users andshould operate at full capacity. One can build an upstream quantumaccess network in which the transmitters are placed at the end userlocation and a common receiver is placed at the network node. This way,an operation with up to 64 users is feasible, which can be done withmulti-user quantum key distribution over a 1×64 passive opticalsplitter.

The above described QKD scheme is applicable to twisted pair, coaxialcable, fiber optic, RF satellite, RF broadcast, RF point-to point, RFpoint-to-multipoint, RF point-to-point (backhaul), RF point-to-point(fronthaul to provide higher throughput CPRI interface forcloudification and virtualization of RAN and cloudified HetNet),free-space optics (FSO), Internet of Things (TOT), Wifi, Bluetooth, as apersonal device cable replacement, RF and FSO hybrid system, Radar,electromagnetic tags and all types of wireless access. The method andsystem are compatible with many current and future multiple accesssystems, including EV-DO, UMB, WIMAX, WCDMA (with or without),multimedia broadcast multicast service (MBMS)/multiple input multipleoutput (MIMO), HSPA evolution, and LTE. The techniques would be usefulfor combating denial of service attacks by routing communications viaalternate links in case of disruption, as a technique to combat TrojanHorse attacks which does not require physical access to the endpointsand as a technique to combat faked-state attacks, phase remappingattacks and time-shift attacks.

Thus, using various configurations of the above described orbitalangular momentum processing, multi-layer overlay modulation, and quantumkey distribution within various types of communication networks and moreparticularly optical fiber networks and free-space optic communicationnetwork, a variety of benefits and improvements in system bandwidth andcapacity maybe achieved.

OAM Based Networking Functions

In addition to the potential applications for static point to point datatransmission, the unique way front structure of OAM beams may alsoenable some networking functions by manipulating the phase usingreconfigurable spatial light modulators (SLMs) or other light projectingtechnologies.

Data Swapping

Data exchange is a useful function in an OAM-based communication system.A pair of data channels on different OAM states can exchange their datain a simple manner with the assistance of a reflective phase hologram asillustrated in FIG. 109. If two OAM beams 10902, 10904, e.g., OAM beamswith l=+L₁ and +L₂, which carry two independent data streams 10906,10908, are launched onto a reflective SLM 10910 loaded with a spiralphase pattern with an order of −(L1+L2), the data streams will swapbetween the two OAM channels. The phase profile of the SLM will changethese two OAM beams to l=−l₂ and l=−l₁, respectively. In addition, eachOAM beam will change to its opposite charge under the reflection effect.As a result, the channel on l=+l₁ is switched to l=+l₂ and vice versa,which indicates that the data on the two OAM channels is exchanged. FIG.109 shows the data exchange between l=+6 10912 and l=8 10914 using aphase pattern on the order of l=−14 on a reflective SLM 10910. A powerpenalty of approximately 0.9 dB is observed when demonstrating this inthe experiment.

An experiment further demonstrated that the selected data swappingfunction can handle more than two channels. Among multiple multiplexedOAM beams, any two OAM beams can be selected to swap their data withoutaffecting the other channels. In general, reconfigurable opticaladd/drop multiplexers (ROADM) are important function blocks in WDMnetworks. A WDM RODAM is able to selectively drop a given wavelengthchannel and add in a different channel at the same wavelength withouthaving to detect all pass-through channels. A similar scheme can beimplemented in an OAM multiplexed system to selectively drop and add adata channel carried on a given OAM beam. One approach to achieve thisfunction is based on the fact that OAM beams generally have a distinctintensity profile when compared to a fundamental Gaussian beam.

Referring now to FIGS. 110 and 111 there is illustrated the manner forusing a ROADM for exchanging data channels. The example of FIG. 111illustrates SLM's and spatial filters. The principle of an OAM-basedROADM uses three stages: down conversion, add/drop and up conversion.The down conversion stage transforms at step 11002 the input multiplexedOAM modes 11102 (donut like transverse intensity profiles 11104) into aGaussian light beam with l=0 (a spotlight transverse intensity profile11106). After the down conversion at step 11002, the selected OAM beambecomes a Gaussian beam while the other beams remain OAM but have adifferent l state. The down converted beams 11106 are reflected at step11004 by a specially designed phase pattern 11108 that has differentgratings in the center and in the outer ring region. The central andouter regions are used to redirect the Gaussian beam 11106 in the center(containing the drop channel 11110) and the OAM beams with a ring-shaped(containing the pass-through channels) in different directions.Meanwhile, another Gaussian beam 11112 carrying a new data stream can beadded to the pass-through OAM beams (i.e., add channel). Following theselective manipulation, an up conversion process is used at step 11006for transforming the Gaussian beam back to an OAM beam. This processrecovers the l states of all of the beams. FIG. 92 illustrates theimages of each step in the add/drop of a channel carried by an OAM beamwith l=+2. Some other networking functions in OAM based systems havealso been demonstrated including multicasting, 2 by 2 switching,polarization switching and mode filtering.

In its fundamental form, a beam carrying OAM has a helical phase frontthat creates orthogonality and hence is distinguishable from other OAMstates. Although other mode groups (e.g., Hermite-Gaussian modes, etc.)also have orthogonality and can be used for mode multiplexing, OAM hasthe convenient advantage of its circular symmetry which is matched tothe geometry of most optical systems. Indeed, many free-space data linkdemonstrations attempt to use OAM-carrying modes since such modes havecircular symmetry and tend to be compatible with commercially availableoptical components. Therefore, one can consider that OAM is used more asa technical convenience for efficient multiplexing than as a necessarily“better” type of modal set.

The use of OAM multiplexing in fiber is potentially attractive. In aregular few mode fiber, hybrid polarized OAM modes can be considered asfiber eigenmodes. Therefore, OAM modes normally have less temporalspreading as compared to LP mode basis, which comprise two eigenmodecomponents each with a different propagation constant. As for thespecially designed novel fiber that can stably propagate multiple OAMstates, potential benefits could include lower receiver complexity sincethe MIMO DSP is not required. Progress can be found in developingvarious types of fiber that are suitable for OAM mode transmission.Recently demonstrated novel fibers can support up to 16 OAM states.Although they are still in the early stages, there is the possibilitythat further improvement of performance (i.e., larger number of“maintained” modes and lower power loss) will be achieved.

OAM multiplexing can be useful for communications in RF communicationsin a different way than the traditional spatial multiplexing. For atraditional spatial multiplexing system, multiple spatially separatedtransmitter and receiver aperture pairs are adopted for the transmissionof multiple data streams. As each of the antenna elements receives adifferent superposition of the different transmitted signals, each ofthe original channels can be demultiplexed through the use of electronicdigital signal processing. The distinction of each channel relies on thespatial position of each antenna pair. However, OAM multiplexing isimplemented such that the multiplexed beams are completely coaxialthroughout the transmission medium, and only one transmitter andreceiver aperture (although with certain minimum aperture sizes) isused. Due to the OAM beam orthogonality provided by the helical phasefront, efficient demultiplexing can be achieved without the assist offurther digital signal post-processing to cancel channel interference.

Many of the demonstrated communication systems with OAM multiplexing usebulky and expensive components that are not necessarily optimized forOAM operation. As was the case for many previous advances in opticalcommunications, the future of OAM would greatly benefit from advances inthe enabling devices and subsystems (e.g., transmitters,(de)multiplexers and receivers). Particularly with regard tointegration, this represents significant opportunity to reduce cost andsize and to also increase performance.

Orthogonal beams using for example OAM, Hermite Gaussian, LaguerreGaussian, spatial Bessel, Prolate spheroidal or other types oforthogonal functions may be multiplexed together to increase the amountof information transmitted over a single communications link. Thestructure for multiplexing the beams together may use a number ofdifferent components. Examples of these include spatial light modulators(SLMs); micro electromechanical systems (MEMs); digital light processers(DLPs); amplitude masks; phase masks; spiral phase plates; Fresnel zoneplates; spiral zone plates; spiral phase plates and phase plates.

Multiplexing Using Holograms

Referring now to FIG. 113, there is illustrated a configuration ofgeneration circuitry for the generation of an OAM twisted beam using ahologram within a micro-electrical mechanical device. Configurationssuch as this may be used for multiplexing multiple OAM twisted beamstogether. A laser 11302 generates a beam having a wavelength ofapproximately 543 nm. This beam is focused through a telescope 11304 andlens 11306 onto a mirror/system of mirrors 11308. The beam is reflectedfrom the mirrors 11308 into a DMD 11310. The DMD 11310 has programmed into its memory a one or more forked holograms 11312 that generate adesired OAM twisted beam 11313 having any desired information encodedinto the OAM modes of the beam that is detected by a CCD 11314. Theholograms 11312 are loaded into the memory of the DMD 11310 anddisplayed as a static image. In the case of 1024×768 DMD array, theimages must comprise 1024 by 768 images. The control software of the DMD11310 converts the holograms into .bmp files. The holograms may bedisplayed singly or as multiple holograms displayed together in order tomultiplex particular OAM modes onto a single beam. The manner ofgenerating the hologram 11312 within the DMD 11310 may be implemented ina number of fashions that provide qualitative differences between thegenerated OAM beam 11313. Phase and amplitude information may be encodedinto a beam by modulating the position and width of a binary amplitudegrating used as a hologram. By realizing such holograms on a DMD thecreation of HG modes, LG modes, OAM vortex mode or any angular mode maybe realized. Furthermore, by performing switching of the generated modesat a very high speed, information may be encoded within the helicity'sthat are dynamically changing to provide a new type of helicitymodulation. Spatial modes may be generated by loading computer-generatedholograms onto a DMD. These holograms can be created by modulating agrating function with 20 micro mirrors per each period.

Rather than just generating an OAM beam 11313 having only a single OAMvalue included therein, multiple OAM values may be multiplexed into theOAM beam in a variety of manners as described herein below. The use ofmultiple OAM values allows for the incorporation of differentinformation into the light beam. Programmable structured light providedby the DLP allows for the projection of custom and adaptable patterns.These patterns may be programmed into the memory of the DLP and used forimparting different information through the light beam. Furthermore, ifthese patterns are clocked dynamically a modulation scheme may becreated where the information is encoded in the helicities of thestructured beams.

Referring now to FIG. 114, rather than just having the laser beam 11402shine on a single hologram multiple holograms 11404 may be generated bythe DMD 4410. FIG. 114 illustrates an implementation wherein a 4×3 arrayof holograms 11404 are generated by the DMD 4410. The holograms 11404are square and each edge of a hologram lines up with an edge of anadjacent hologram to create the 4×3 array. The OAM values provided byeach of the holograms 11404 are multiplexed together by shining the beam11402 onto the array of holograms 11404. Several configurations of theholograms 11404 may be used in order to provide differing qualities ofthe OAM beam 11313 and associated modes generated by passing a lightbeam through the array of holograms 11404.

Referring now to FIG. 115 there is illustrated an alternative way ofmultiplexing various OAM modes together. An X by Y array of holograms11502 has each of the hologram 11502 placed upon a black (dark)background 11504 in order to segregate the various modes from eachother. In another configuration illustrated in FIG. 116, the holograms11602 are placed in a hexagonal configuration with the background in theoff (black) state in order to better segregate the modes.

FIG. 117 illustrates yet another technique for multiplexing multiple OAMmodes together wherein the holograms 11702 are cycled through in a loopsequence by the DMD 11310. In this example modes T₀-T₁₁ are cycledthrough and the process repeats by returning back to mode T₀. Thisprocess repeats in a continuous loop in order to provide an OAM twistedbeam with each of the modes multiplex therein.

In addition to providing integer OAM modes using holograms within theDMD, fractional OAM modes may also be presented by the DMD usingfractional binary forks as illustrated in FIG. 118. FIG. 118 illustratesfractional binary forks for generating fractional OAM modes of 0.25,0.50, 0.75, 1.25, 1.50 and 1.75 with a light beam.

Referring now to FIG. 119-132, there are illustrated the resultsachieved from various configurations of holograms program within thememory of a DMD. FIG. 119 illustrates the configuration at 11902 havingno hologram separation on a white background producing the OAM modeimage 11904. FIG. 120 uses a configuration 12002 consisting of circularholograms 11904 having separation on a white background. The OAM modeimage 11906 that is provided therefrom is also illustrated. Bright modeseparation yields less light and better mode separation.

FIG. 121 illustrates a configuration 12102 having square holograms withno separation on a black background. The configuration 12102 generatesthe OAM mode image 12104. FIG. 122 illustrates the configuration ofcircular holograms (radius ˜256 pixels) that are separated on a blackbackground. This yields the OAM mode image 12204. Dark mode separationyields more light in the OAM image 12204 and has slightly better modeseparation.

FIG. 123 illustrates a configuration 12302 having a bright backgroundand circular hologram (radius ˜256 pixels) separation yielding an OAMmode image 12304. FIG. 124 illustrates a configuration 12402 usingcircular holograms (radius ˜256 pixels) having separation on a blackbackground to yield the OAM mode image 12404. The dark mode separationyields more light and has a slightly worse mode separation within theOAM mode images.

FIG. 125 illustrates a configuration 12502 including circular holograms(radius ˜256 pixels) in a hexagonal distribution on a bright backgroundyielding an OAM mode image 12504. FIG. 126 illustrates at 12602 smallcircular holograms (radius ˜256 pixels) in a hexagonal distribution on abright background that yields and OAM mode image 12604. The largerholograms with brighter backgrounds yield better OAM mode separationimages.

Referring now to FIG. 127, there is illustrated a configuration 12702 ofcircular holograms (radius ˜256 pixels) in a hexagonal distribution on adark background with each of the holograms having a radius ofapproximately 256 pixels. This configuration 12702 yields the OAM modeimage 12704. FIG. 128 illustrates the use of small holograms (radius˜256 pixels) having a radius of approximately 190 pixels arranged in ahexagonal distribution on a black background that yields the OAM modeimage 12804. Larger holograms (radius of approximately 256 pixels)having a dark background yields worse OAM mode separation within the OAMmode images.

FIG. 129 illustrates a configuration 12902 of small holograms (radius ofapproximately 190 pixels) in a hexagonal separated distribution on adark background that yields the OAM mode image 12904. FIG. 130illustrates a configuration 13002 of small holograms (radius ˜256pixels) in a hexagonal distribution that are close together on a darkbackground that yields the OAM mode image 13004. The larger darkboundaries (FIG. 129) yield worse OAM mode image separation than asmaller dark boundary.

FIG. 131 illustrates a configuration 13102 of small holograms (radius˜256 pixels) in a separated hexagonal configuration on a brightbackground yielding OAM mode image 13104. FIG. 63 illustrates aconfiguration 6302 of small holograms (radius ˜256 pixels) more closelyspaced in a hexagonal configuration on a bright background yielding OAMmode image 6304. The larger bright boundaries (FIG. 131) yield a betterOAM mode separation.

Additional illustrations of holograms, namely reduced binary hologramsare illustrated in FIGS. 133-136. FIG. 133 illustrates reduced binaryholograms having a radius equal to 100 micro mirrors and a period of 50for various OAM modes. Similarly, OAM modes are illustrated for reducedbinary for holograms having a radius of 50 micro mirrors and a period of50 (FIG. 134); a radius of 100 micro mirrors and a period of 100 (FIG.135) and a radius of 50 micro mirrors and a period of 50 (FIG. 136).

The illustrated data with respect to the holograms of FIGS. 119-136demonstrates that full forked gratings yield a great deal of scatteredlight. Finer forked gratings yield better define modes within OAMimages. By removing unnecessary light from the hologram (white regions)there is a reduction in scatter. Holograms that are larger and havefewer features (more dark zones) having a hologram diameter of 200 micromirrors provide overlapping modes and strong intensity. Similarconfigurations using 100 micro mirrors also demonstrate overlappingmodes and strong intensity. Smaller holograms having smaller radiibetween 100-200 micro mirrors and periods between 50 and 100 generatedby a DLP produce better defined modes and have stronger intensity thanlarger holograms with larger radii in periods. Smaller holograms havingmore features (dark zones with hologram diameters of 200 micro mirrorsprovide well-defined modes with strong intensity. However, hundred micromirror diameter holograms while providing well-defined modes provideweaker intensity. Thus, good, compact hologram sizes are between 100-200micro mirrors with zone periods of between 50 and 100. Larger hologramshave been shown to provide a richer OAM topology.

Referring now to FIGS. 139 and 140, there are illustrated a blockdiagram of a circuit for generating a muxed and multiplexed data streamcontaining multiple new Eigen channels (FIG. 139) for transmission overa communications link (free space, fiber, RF, etc.), and a flow diagramof the operation of the circuit (FIG. 140). Multiple data streams 13902are received at step 14002 and input to a modulator circuit 13904. Themodulator circuit 13904 modulates a signal with the data stream at step14004 and outputs these signals to the orthogonal function circuit13906. The orthogonal function circuit 13906 applies a differentorthogonal function to each of the data streams at step 14006. Theseorthogonal functions may comprise orbital angular momentum functions,Hermite Gaussian functions, Laguerre Gaussian functions, prolatespheroidal functions, Bessel functions or any other types of orthogonalfunctions. Each of the data streams having an orthogonal functionapplied thereto are applied to the mux circuit 13098. The mux circuit13098 performs a spatial combination of multiple orthogonal signals ontoa same physical bandwidth at step 14008. Thus, a single signal willinclude multiple orthogonal data streams that are all located within thesame physical bandwidth. A plurality of these muxed signals are appliedto the multiplexing circuit 13910. The multiplexing circuit 13910multiplexes multiple muxed signals onto a same frequency or wavelengthat step 14010. Thus, the multiplexing circuit 13910 temporallymultiplexes multiple signals onto the same frequency or wavelength. Themuxed and multiplexed signal is provided to a transmitter 13912 suchthat the signal 13914 may be transmitted at step 14012 over acommunications link (Fiber, FSO, RF, etc.).

Referring now to FIGS. 141 and 142, there is illustrated a block diagram(FIG. 141) of the receiver side circuitry and a flow diagram (FIG. 142)of the operation of the receiver side circuitry associated with thecircuit of FIG. 139. A received signal 14102 is input to the receiver14104 at step 14202. The receiver 14014 provides the received signal14102 to the de-multiplexer circuit 14106. The de-multiplexer circuit14106 separates the temporally multiplexed received signal 14102 intomultiple muxed signals at step 14204 and provides them to the de-muxcircuit 14108. As discussed previously with respect to FIGS. 139 and140, the de-multiplexer circuit 14106 separates the muxed signals thatare temporally multiplexed onto a same frequency or wavelength. Thede-mux circuit 14108 separates (de-muxes) the multiple orthogonal datastreams at step 14206 from the same physical bandwidth. The multipleorthogonal data streams are provided to the orthogonal function circuit14110 that removes the orthogonal function at step 14208. The individualdata streams may then be demodulated within the demodulator circuit14112 at step 14210 and the multiple data streams 14114 provided foruse.

Thus, the above described process enables multiple data streams to befirst placed within a same physical bandwidth to create a muxed signalof orthogonal data streams. Multiple of these muxed signals may then bemultiplexed onto a same frequency or wavelength in order to provide moreinformation on a same communications link. Each orthogonal functionwithin the muxed signals that are then multiplexed together represents anew Eigen channels that may carry a unique information stream thusgreatly increasing the amount of data which may be transmitted over thecommunications link. As described here and above the communications linkmay comprise free space optical, fiber, RF or any other communicationstructure. The manner for muxing and multiplexing the data may also useany of the processing techniques described herein above.

Optical communications may be carried out over an optical fiber usingoptical signals processed with orthogonal functions such asHermite-Gaussian functions and Laguerre-Gaussian functions as describedhereinabove in order to improve system bandwidth. Laguerre-Gaussian (LG)and Hermite-Gaussian (HG) signals have three important properties andadvantages and one major disadvantage. The advantages include theability to form two complete families of exact and orthogonal solutionsof the paraxial wave equations. Another advantage is that the HG and LGsignals are transverse eigenmodes of stable resonators. Finally, the HGand LG signals do not change shape on propagation and provide stablemodes of propagation for signals. The disadvantage of HG and LG signalsis that the eigenmodes coupled to one another after a long distance ofpropagation. However, for short distance applications such as intra-dataand inter-data center connectivity, and front haul applications or backhaul applications, both LG and HG signals provide good solutions withinthese applications.

Referring now to FIG. 143, there are illustrated various types oforthogonal functions that may be utilized in optical fibertransmissions. Hermite-Gaussian functions 14302 use Hermite-Gaussian(HG) signals that are derived from the expansion of Laplacian equationsin a rectangular coordinate system 14304 with paraxial approximation inthe z-direction. Laguerre-Gaussian functions 14306 use Laguerre-Gaussian(LG) signals that are derived from the expansion of Laplacian equationsin a cylindrical coordinate system with a paraxial approximation in theZ-direction. Depending on the geometrical symmetries of the problem, forexample, one mode would propagate better than the others if they are thenatural symmetry of the chosen geometry. Both HG and LG signals havebeen discussed in corresponding U.S. patent application Ser. No.14/882,085, filed Oct. 13, 2015, entitled APPLICATION OF ORBITAL ANGULARMOMENTUM TO FIBER, FSO AND RF, which is incorporated herein by referencein its entirety. The theoretical connection between the two modes hasalso been studied on a Poincare sphere in U.S. patent application Ser.No. 14/818,050, entitled MODULATION AND MULTIPLE ACCESS TECHNIQUE USINGORBITAL ANGULAR MOMENTUM, which is incorporated herein by reference inits entirety.

A technique for further improving system bandwidth in optical fibertransmissions involves the use of input signals processed using anelliptical coordinate system that is transmitted over an elliptical corefiber. Elliptical core fibers may be used with Ince-Gaussian spatialorthogonal wave fronts for spatial modulation. Ince-Gaussian functions14310 use Ince-Gaussian (IG) signals that are derived from the expansionof Laplacian equation in an elliptical coordinate system with paraxialapproximation in the Z-direction. The Ince-Gaussian functions 14310 maybe applied in each of the same manners and applications as theHermite-Gaussian functions, Laguerre-Gaussian functions and otherorthogonal functions describe previously herein.

Ince-Gaussian signals comprise a third family of exacting orthogonalsolutions of the paraxial wave equation in the elliptical coordinatesystem. These new Eigen functions have natural resonating modes instable resonators and constitute the exact and continuous transitionmodes between HG and LG eigenmodes. The transverse distribution of thesefields is described by new Eigen modes and HG and LG have a complete setof solutions of the paraxial wave equations, such that any paraxialfield can be expressed as a superposition of these eigenmodes. Thegeneration of multiple modes of these Eigen functions, the muxing ofthem, their demuxing, and detection characteristics of the eigenmodeshave potential uses of such modes in a commercial setting. These neweigenmodes can approximate LG or HG as a parameter of the Wignertransform going to zero or infinity.

Referring now to FIG. 144, there is illustrated a transmitter 14402 andreceiver 14404 transmitting signals over an elliptical core fiber 14406.In a fiber 14406 with an elliptical core, such elliptical eigenmodes, asprovided by Ince-Gaussian functions, can propagate further beforecoupling with one another and more of these modes can be muxed togetherwithin an elliptical core fiber as compared to other modes such as LGmodes or HG modes. A passively specially designed lens 14408 having adiamond cut such that plane waves 14410 incoming at different angles canbe converted to elliptical modes 14412 and muxed together before leavingthe lens 14408. The elliptical modes 14412 can be demuxed with anotherspecially designed lens 14414 that demultiplexes and converts theelliptical modes 14412 back into plane waves 14410. Thus, eachorthogonal eigenmode acts as an independent channel on one wavelengthover the elliptical fiber 14406. Therefore, techniques such as CWDM(course wavelength division multiplexing) or DWDM (dense wavelengthdivision multiplexing) are still applicable as such eigenmodes are muxedon one particular wavelength and therefore, can create multipleindependent channels on one specific wavelength and more wavelengths canbe aggregated together to increase throughput or capacity of the system.

Referring now more particularly to FIG. 145, there is generallyillustrated the process for generating an Ince-Gaussian process signalfor transmission on elliptical fiber. An input signal/signals 14502comprises the data or information that is to be transmitted to theoptical processing circuitry 14504 for processing. The opticalprocessing circuitry 14504, in one example, comprises the circuitrydescribed in corresponding U.S. patent application Ser. No. 14/882,085.While the disclosure of the '085 application describes applying eitherrectangular coordinates that yield Hermite-Gaussian signals orcylindrical coordinates providing Laguerre-Gaussian signals, the opticalprocessing circuitry 1504 of the present system processes the inputsignals 15402 into an optical signal using an elliptical coordinatesystem.

The optical processing circuitry 14504 expands the Laplacian of the waveequation in elliptical contexts and solves them as separable solutionsfor each coordinate. The angular solutions produce new functions calledMathieu and modified Mathieu functions. The optical processing circuitry14504 performs a paraxial approximation to produce new differentialequations and solutions for the paraxial equations. The process signalsmay be generated using spatial light modulators (SLM's) or othertechniques such as those described in U.S. patent application Ser. No.14/882,085 to transmit the information over an elliptical core fiber14508. The optical processing circuitry 14504 may perform muxing andmultiplexing process to combine IG processed signals. A multiplexingtechnique implemented by the optical processing circuitry 14504 cansequentially combine different Ince Gaussian orthogonal modes in anelliptical core fiber with a higher multiplier effect that achieves ahigher number of orthogonal Eigen channels with distinct modalcombinations.

Ince-Gaussian (IG) beams generated within the optical processingcircuitry 14504 are the solutions of paraxial beams in an ellipticalcoordinate system. IG beams are the third kind of orthogonal Eigenstates and can probe the chirality structures of samples. Since IG modeshave a preferred symmetry (long axis versus short axis) this enables itto probe chirality better than Laguerre-Gaussian or Hermite-Gaussianmodes. This enables the propagation of more IG modes within anelliptical core fiber than Laguerre-Gaussian modes or Hermite-Gaussianmodes.

The solution of the Ince-Gaussian equations is performed in anelliptical coordinate system. The wave equation can be represented as aHelmholtz equation in Cartesian coordinates as follows:

(∇² +k ²)E(x,y,z)=0

E(x,y, z) is complex field amplitude which can be expressed in terms ofits slowly varying envelope and fast varying part in z-direction.

E(x,y,z)=ψ(x,y,z)e ^(jkz)

A Paraxial Wave approximation may be determined by substituting ourassumption in the Helmholtz Equation.

(∇²+k²)ψ ⋅ e^(jkz) = 0${\frac{\delta^{2}\psi}{\delta \; x^{2}} + \frac{\delta^{2}\psi}{\delta \; y^{2}} + \frac{\delta^{2}\psi}{\delta \; z^{2}} - {j\; 2k\frac{\delta\psi}{\delta \; z}}} = 0$

A slowly varying envelope approximation is made as follows:

${{\frac{\delta^{2}\psi}{\delta \; z^{2}}}{\frac{\delta^{2}\psi}{\delta \; x^{2}}}},{\frac{\delta^{2}\psi}{\delta \; y^{2}}},{2k{\frac{\delta\psi}{\delta \; z}}}$${{\nabla_{t}^{2}\psi} + {j\; 2k\frac{\delta\psi}{\delta \; z}}} = 0$

This comprises a Paraxial wave equation.

The elliptical-cylindrical coordinate system is defined as shown in FIG.146.

x = acos h ξcosη y = a sin  h ξsinη ξ ∈ (0, ∞), η ∈ (0, 2π)$a = {{{f(z)}\mspace{14mu} {where}\mspace{14mu} {f(z)}} = \frac{f_{0}{w(z)}}{w_{0}}}$

Curves of constant value of ξ trace confocal ellipses as shown in FIG.147.

${\frac{x^{2}}{a^{2}\cos^{2}\xi} + \frac{y^{2}}{a^{2}\sinh^{2}\xi}} = {1\mspace{14mu} ({Ellipse})}$

A constant value of η give confocal hyperbolas as shown in FIG. 148.

${\frac{x^{2}}{a^{2}\cos^{2}\eta} - \frac{y^{2}}{a^{2}\sin^{2}\eta}} = {1\mspace{14mu} ({hyperbola})}$

An elliptical-cylindrical coordinate system may then be defined in thefollowing manner:

$\nabla_{t}^{2}{= {{\frac{1}{h_{\xi}^{2}}\frac{\delta^{2}}{{\delta\xi}^{2}}} + {\frac{1}{h_{\eta}^{2}}\frac{\delta^{2}}{{\delta\eta}^{2}}}}}$

where h_(ξ), h_(η) are scale factors

$h_{\xi} = \sqrt{\left( \frac{\delta \; x}{\delta\xi} \right)^{2} + \left( \frac{\delta \; y}{\delta\xi} \right)^{2}}$$h_{\eta} = \sqrt{\left( \frac{\delta \; x}{\delta\eta} \right)^{2} + \left( \frac{\delta \; y}{\delta\eta} \right)^{2}}$$h_{\xi} = {h_{\eta} = {a\sqrt{{\sinh^{2}\xi} + {\sin^{2}\eta}}}}$$\nabla_{t}^{2}{= {\frac{1}{a^{2}\sinh^{2}{\xi sin}^{2}\eta}\left( {\frac{\delta^{2}}{{\delta\xi}^{2}} + \frac{\delta^{2}}{{\delta\eta}^{2}}} \right)}}$

The solution to the paraxial wave equations may then be made inelliptical coordinates. Paraxial Wave Equation in Elliptic Cylindricalco-ordinates are defined as:

${{\frac{1}{a^{2}\left( {\sinh^{2}{\xi sin}^{2}\eta} \right)}\left( {\frac{\delta^{2}\psi}{{\delta\xi}^{2}} + \frac{\delta^{2}\psi}{{\delta\eta}^{2}}} \right)} - {j\; 2k\frac{\delta\psi}{\delta \; z}}} = 0$

Assuming separable solution as modulated version of fundamental Gaussianbeam.

${{IG}\left( \overset{\sim}{r} \right)} = {{E(\xi)}{N(\eta)}{\exp \left( {{jZ}(z)} \right)}{\psi_{GB}\left( \overset{\sim}{r} \right)}}$${{where}\mspace{14mu} {\psi_{GB}\left( \overset{\sim}{r} \right)}} = {\frac{w_{0}}{w(z)}{\exp \left\lbrack {{- \frac{r^{2}}{w^{2}(z)}} + {j\frac{{kr}^{2}}{2{R(z)}}} - {j\; {\psi_{GS}(z)}}} \right\rbrack}}$

E, N & Z are real functions. They have the same wave-fronts as ψ_(GB)but different intensity distribution.

Separated differential equations are defined as:

${\frac{d^{2}E}{d\; \xi^{2}} - {{\varepsilon sinh2\xi}\frac{dE}{d\; \xi}} - {\left( {a - {p\; {\varepsilon cosh2\xi}}} \right)E}} = 0$${\frac{d^{2}N}{d\; \eta^{2}} - {{\varepsilon sin2\eta}\frac{dN}{d\; \eta}} - \left( {a - {p\; {\varepsilon cos2\eta}}} \right)} = {{0 - {\left( \frac{z^{2} + z_{r}^{2}}{z_{r}} \right)\frac{dZ}{dz}}} = p}$

where a and p are separation constants

$\varepsilon = \frac{f_{0}w_{0}}{w(z)}$

The even solutions for the Ince-Gaussian equations are:

${{IG}_{pm}^{e}\left( {\overset{\sim}{r},\varepsilon} \right)} = {\frac{{Cw}_{o}}{w(z)}{C_{p}^{m}\left( {{j\; \xi},\varepsilon} \right)}{C_{p}^{m}\left( {\eta,\varepsilon} \right)}{\exp \left( {- \frac{r^{2}}{w^{2}(z)}} \right)} \times \exp \; {j\left( {{kz} + \frac{{kr}^{2}}{2{R(z)}} - {\left( {p + 1} \right){\psi_{GS}(z)}}} \right)}}$

The frequency of the even Ince-Polynomials are illustrated in FIGS. 149Aand 149B and the modes and their phases are illustrated in FIG. 150.

The odd solutions for the Ince-Gaussian equations are:

${{IG}_{pm}^{o}\left( {\overset{\sim}{r},\varepsilon} \right)} = {\frac{{sw}_{0}}{w(z)}{S_{p}^{m}\left( {{j\; \xi},\varepsilon} \right)}{S_{p}^{m}\left( {\eta,\varepsilon} \right)}{\exp \left( {- \frac{r^{2}}{w^{2}(z)}} \right)} \times \exp \; {j\left( {{kz} + \frac{{kr}^{2}}{2{R(z)}} - {\left( {p + 1} \right){\psi_{GS}(z)}}} \right)}}$

The frequency of the odd Ince Polynomials are illustrated in FIGS. 151Aand 151B and the modes and their phases are illustrated in FIG. 152.Processing of the input signal 14502 by the optical processing circuitry14504 using an elliptical coordinate system creates a unique itemfunction that provides improved throughput characteristics when utilizedupon an elliptical core optical fiber 14508 when transmitted from anoptical transmitter 14506 utilizing the processed signal.

Referring now back to FIG. 145, the optical transmitter 14506 transmitsthe generated Ince-Gaussian beam over the elliptical fiber 14508. Thesignals processed according to the elliptical coordinate system willprovide better transmission characteristics over the elliptical coreoptical fiber 14508 allowing greater data throughput. The input signal14502 processed according to the elliptical core system will have adistinct index of refraction profile arising from processing using theelliptical coordinate system. The signals transmitted over theelliptical fiber 14508 are processed and demodulated using a receiveroptical processing circuitry 14510. The receiver optical processingcircuitry 14510 removes the Eigen function applied using the ellipticalcoordinate system to reconstruct an output signal 14512 that comprisesthe originally received input signals.

In an alternative embodiment, the transmitter optical processingcircuitry 14504 and the receiver optical processing circuitry 14510 mayutilize the techniques discussed in U.S. patent application Ser. No.14/882,085 to generate the Laguerre-Gaussian, Hermite-Gaussian orInce-Gaussian processed signals that are transmitted over an ellipticalcore fiber to improve data bandwidth.

Referring now to FIG. 153, the elliptical core fiber 14508 has aparabolic index profile and is weakly guided. These are boundaryconditions that are input to the wave equation (Laplacian). Also, theelliptical fiber 14508 includes stress rods 15304 on either size of thesemi-major axis (the longer elliptical axis) 15305 along the semi-minoraxis (the shorter elliptical axis) 15306 that may be taken into accountin the Laplacian wave equation. The elliptical fiber 14508 may beproduced using the preform techniques similar to those disclosed in U.S.patent application Ser. No. 15/430,981, filed Feb. 13, 2017, entitledSYSTEM AND METHOD FOR PRODUCING VORTEX FIBER, which is incorporatedherein by reference in its entirety or using the fiber formingtechniques disclosed in U.S. patent application Ser. No. 14/882,085,filed Oct. 13, 2015, entitled APPLICATION OF ORBITAL ANGULAR MOMENTUM TOFIBER, FSO AND RF, which is incorporated herein by reference in itsentirety.

An elliptical core fiber may comprise a few mode fiber as illustrated inFIG. 154. The elliptical core-few mode fiber (EC-FMF) 15402 includes anelliptical core 15404 within the center of the fiber that has a majoraxis of approximately 15 μm and a minor axis of approximately 10 μm. Thecore shape is described by the ovality parameter:

$o = \frac{2\left( {a - b} \right)}{a + b}$

Here,

a=15 μm

b=10 μm

Giving ovality,

o=40%

The two dark regions 15406 along the minor axis are stress-applyingparts (stress and rods) for creating asymmetric stress during the fiberdraw process to make the core elliptical.

Ovality may also be written o→X=2(R_(a)−R_(b))/(R_(a)+R_(b)) and is ameasure of how “out of round” cylindrical part is.

Referring now to FIG. 155, there are illustrated intensity diagrams forvarious different types of beam topologies within elliptical corefibers. Row 15502 illustrates linearly polarized signals transmittedthrough an elliptical core fiber for various modes. Row 15504illustrates intensity diagrams for Laguerre-Gaussian signals for LG₀₁mode, LG₁₀ mode and LG₀₁ mode. Row 15506 illustrates the intensitydiagrams for Hermite-Gaussian signals for HG₀₁ mode, HG₁₀ mode and HG₂₀mode. Finally, row 15508 illustrates the intensity diagrams forInce-Gaussian beams in IG₁₁ mode, IG₂₀ mode and IG₂₂ mode.

A mode crosstalk matrix provides an illustration of the relative amountof light scattered into adjacent modes that were not launched into afiber. Referring now to FIG. 156 there is illustrated a measurementtechnique for generating a mode crosstalk matrix. FIG. 157 is a flowdiagram illustrating the steps of the process. A 980 nm laser 15602generates a laser signal that is transmitted through a first lens 156042A first spatial light modulator 15606 special modulator 15606 utilizes ahologram 15608 to generate a first Ince-Gaussian beam having aparticular intensity diagram 15610. The generated Ince-Gaussian beamfrom the SLM 15606 passes through a series of lenses 15612 that focusesand launches the generated Ince-Gaussian beam having a particular modeonto a elliptical core few mode fiber 15614 at step 15702 (see FIG.157).

After passing through the fiber 15614, the output of the fiber ismonitored at step 15704 and will comprise a superposition of the variousoutput mode components. The output signal passes through a lens 15616 toa second spatial light modulator (SLM) 15618. For every output modecomponent to be measured, the SLM 15618 will imprint at step 15706 aninverse of the modes face front onto the output beam from the fiber15614. The output of the SLM 15618 next passes through a further seriesof lenses 15620 before being input to a spatial filter 15622. The outputmode components from the SLM 15618 to be measured will have a planarface front after the imprinting with the inverse of the Ince-Gaussianbeam. The spatial filter 15622 passes the plane wave component andblocks other output mode components within the signal from the SLM 15618at step 15710. The relative amount of power through the spatial filter15622 is the amount of the output mode component to be measured at step15712. By interrogating different components at step 15714, the rows ofthe mode crosstalk matrix may be filled in for the fiber under test.After the current mode is complete, a new mode may be launched at step15716 into the fiber under test to complete a next row of the crosstalkmatrix. The above process is repeated for each mode to be tested.

FIG. 158 illustrates a generated single row of the mode crosstalkmatrix. The mode HG₁₁ has been launched into the fiber 15614. The poweroutput is measured for various other modes within the row. Thus, theselectively measured output for mode HG₀₀ is −23.7, for HG₀₁ is −21.2,for HG₁₀ is −18.8, for HG₀₂ is −17.1 and for HG₂₀ is −20.9.

Referring now to FIG. 159, there is illustrated the results for thecomparison of each of the selectively excited modes 15902 that arecalculated from the first SLM 15606. Each of the modes after passingthrough the fiber 15614 is measured by set of masks (holograms) 15904using the second SLM 15618 to yield an associated set of intensityimages 15906. For each of the intensity images 15906, the optical powerof the light coupled into the fiber is measured. These are representedfor the various modes and a level from −28.1 dB to −14.5 dB is indicatedgenerally at 15908.

Each of the selectively measured outputs 15908 that are measured may beused to populate a mode crosstalk matrix using Hermite-Gaussian modes asillustrated in FIG. 160. The mode crosstalk matrix 16002 generates thepower values for provided input values according to dB→10 log₁₀P_(ij)/P_(ii) in each row and are measured for each selectively measuredoutput by the spatial light modulator 15618 for a given selectivelyexcited input to the fiber 15614 by the SLM 15606. The rows of the modecrosstalk matrix 16002 comprise the modes selectively excited by thespatial light modulator 15606. The columns of the mode crosstalk matrix16002 comprise the modes selectively excited by the spatial lightmodulator 15618. A mode crosstalk matrix is for Laguerre-Gaussian modesand linearly polarized modes are illustrated in FIGS. 161 and 162,respectively.

In addition to those various orthogonal function mode techniquesdiscussed hereinabove to which Ince Gaussian functions may be applied ina manner similar to that of Hermite Gaussian, Laguerre Gaussian or othertypes of orthogonal functions, Ince Gaussian orthogonal modes/functionsmay be applied in a number of other manners. These include the use ofInce Gaussian orthogonal modes in elliptical core fibers to performphase estimation in carer recovery, to perform symbol. Estimation andclock recovery, to perform decision directed carrier recovery wheredecoded signals are compared with the closest constellation points orfor dealing with amplifier nonlinearity. Ince Gaussian orthogonal modeswithin an elliptical core fiber may also be used for adaptive powercontrol, adaptive variable symbol rate and within adaptive equalizationtechniques.

It will be appreciated by those skilled in the art having the benefit ofthis disclosure that this system and method for transmissions usingelliptical core fibers provides an improved method for optical signaltransmission within a fiber. It should be understood that the drawingsand detailed description herein are to be regarded in an illustrativerather than a restrictive manner, and are not intended to be limiting tothe particular forms and examples disclosed. On the contrary, includedare any further modifications, changes, rearrangements, substitutions,alternatives, design choices, and embodiments apparent to those ofordinary skill in the art, without departing from the spirit and scopehereof, as defined by the following claims. Thus, it is intended thatthe following claims be interpreted to embrace all such furthermodifications, changes, rearrangements, substitutions, alternatives,design choices, and embodiments.

What is claimed is:
 1. A system for transmission of optical datasignals, comprising: first optical processing circuitry for receiving aplurality of digital signals and applying at least one of aHermite-Gaussian function, Laguerre-Gaussian function or anInce-Gaussian function to each of the received plurality of digitalsignals and combining each of the at least one of Hermite-Gaussian,Laguerre-Gaussian or Ince-Gaussian function applied plurality of digitalsignals into a single carrier signal; an optical transmitter fortransmitting the single carrier signal; an optical receiver forreceiving the transmitted single carrier signal; second opticalprocessing circuitry for separating the at least one of theHermite-Gaussian function, the Laguerre-Gaussian function or theInce-Gaussian function applied digital signals of the single carriessignal into separate signals and for removing the at least one of theHermite-Gaussian function, the Laguerre-Gaussian function or theInce-Gaussian function applied to each of the plurality of digitalsignals; and an elliptical core fiber for transmitting the singlecarrier signal from the optical transmitter to the optical receiver, theelliptical core fiber including an elliptical core have a major axis anda minor axis.
 2. The system of claim 1, wherein the elliptical corefiber further comprises a first stress rod located above the major axisalong the minor axis and a second stress rod located below the majoraxis along the minor axis.
 3. The system of claim 1, wherein the firstoptical processing circuitry further comprises a spatial light modulatorfor generating and applying the at least one of the Hermite-Gaussianfunction, the Laguerre-Gaussian function or the Ince-Gaussian functionto each of the plurality of digital signals.
 4. The system of claim 1,wherein the second optical processing circuitry further comprises asecond spatial light modulator for imprinting an inverse phase front ofeach received mode component on each the received mode components withinthe received single carrier signal to generate the carrier signal havinga planar phase front for each of the mode components within the carriersignal.
 5. The system of claim 4 further including a spatial filter forfiltering the carrier signal from the second spatial light modulator,wherein the spatial filter passes plane wave components of the of thecarrier signal and blocks other output mode components of the carriersignal.
 6. The system of claim 5, wherein the second optical processingcircuitry further measures a power output of each mode component fromthe spatial filter.
 7. The system of claim 6, wherein the measured poweroutput of each mode component is used to populate a mode crosstalkmatrix.
 8. The system of claim 1, wherein the optical transmitterlaunches the single carrier into the elliptical core fiber.
 9. Thesystem of claim 1, wherein the at least one of the Hermite-Gaussianfunction, the Laguerre-Gaussian function or the Ince-Gaussian functioncomprises the Ince-Gaussian function.
 10. The system of claim 1, whereinthe first optical processing circuitry further combines each of the atleast one of Hermite-Gaussian, Laguerre-Gaussian or Ince-Gaussianfunction applied plurality of digital signals into a single carriersignal using at least one of wavelength division multiplexing or densewavelength division multiplexing.
 11. The system of claim 1, wherein thefirst optical processing circuitry further combines each of the at leastone of Hermite-Gaussian, Laguerre-Gaussian or Ince-Gaussian functionapplied plurality of digital signals into a single carrier signal usinga combination of polarization multiplexing, wavelength divisionmultiplexing and mode division multiplexing.
 12. A method fortransmission of optical data signals, comprising: receiving a pluralityof digital signals; applying at least one of a Hermite-Gaussianfunction, a Laguerre-Gaussian function or an Ince-Gaussian function toeach of the received plurality of digital signals; combining each of theat least one of the Hermite-Gaussian function, the Laguerre-Gaussianfunction or the Ince-Gaussian function applied plurality of digitalsignals into a single carrier signal; transmitting the single carriersignal over an elliptical core fiber, the elliptical core fiberincluding an elliptical core have a major axis and a minor axis;receiving the transmitted single carrier signal over the elliptical corefiber; separating the at least one of the Hermite-Gaussian function, theLaguerre-Gaussian function or the Ince-Gaussian function applied digitalsignals of the single carries signal into separate signals; and removingthe at least one of the Hermite-Gaussian function, the Laguerre-Gaussianfunction or the Ince-Gaussian function applied to each of the pluralityof digital signals.
 13. The method of claim 12, wherein the step oftransmitting further comprises transmitting the single carrier signalover an elliptical core fiber comprising a first stress rod locatedabove the major axis of the core of the elliptical core fiber along theminor axis of the core of the elliptical core fiber and a second stressrod located below the major axis of the elliptical core fiber along theminor axis elliptical core fiber.
 14. The method of claim 12 wherein thestep of applying further comprises generating and applying the at leastone of the Hermite-Gaussian function, the Laguerre-Gaussian function orthe Ince-Gaussian function to each of the plurality of digital signalsusing a spatial light modulator.
 15. The method of claim 12, wherein thestep of removing further comprises imprinting an inverse phase front ofeach received mode component on each the received mode components withinthe received single carrier signal to generate the carrier signal havinga planar phase front for each of the mode components within the carriersignal using a second spatial light modulator.
 16. The method of claim15 further including filtering the carrier signal from the secondspatial light modulator using a spatial filter.
 17. The method of claim16 wherein the step of filtering further comprises: passing plane wavecomponents of the of the carrier signal; and blocking other output modecomponents of the carrier signal.
 18. The method of claim 17 furthercomprising measuring a power output of each mode component passed fromthe spatial filter.
 19. The method of claim 18 further including thestep of populating a mode crosstalk matrix with the measured poweroutput of each mode component.
 20. The system of claim 12, wherein thestep of transmitting further comprises launching the single carrier intothe elliptical core fiber.
 21. The method of claim 12, wherein the atleast one of the Hermite-Gaussian function, the Laguerre-Gaussianfunction or the Ince-Gaussian function comprises the Ince-Gaussianfunction.
 22. The method of claim 12, wherein the step of combiningfurther comprises combining each of the at least one ofHermite-Gaussian, Laguerre-Gaussian or Ince-Gaussian function appliedplurality of digital signals into a single carrier signal using at leastone of wavelength division multiplexing or dense wavelength divisionmultiplexing.
 23. The method of claim 12, wherein the step of combiningfurther comprises combining each of the at least one ofHermite-Gaussian, Laguerre-Gaussian or Ince-Gaussian function appliedplurality of digital signals into a single carrier signal using acombination of polarization multiplexing, wavelength divisionmultiplexing and mode division multiplexing.
 24. A system fortransmission of optical data signals, comprising: first opticalprocessing circuitry for receiving a plurality of digital signals andapplying at least one of a Hermite-Gaussian function, aLaguerre-Gaussian function or an Ince-Gaussian function to each of thereceived plurality of digital signals and combining each of the at leastone of the Hermite-Gaussian function, the Laguerre-Gaussian function orthe Ince-Gaussian function applied plurality of digital signals into asingle carrier signal, wherein the first optical processing circuitryfurther comprises a spatial light modulator for generating and applyingthe at least one of the Hermite-Gaussian function, Laguerre-Gaussianfunction or the Ince-Gaussian function to each of the plurality ofdigital signals; an optical transmitter for transmitting the singlecarrier signal; an optical receiver for receiving the transmitted singlecarrier signal; second optical processing circuitry for separating theat least one of the Hermite-Gaussian function, the Laguerre-Gaussianfunction or the Ince-Gaussian function applied digital signals of thesingle carries signal into separate signals and for removing the atleast one of the Hermite-Gaussian function, the Laguerre-Gaussianfunction or the Ince-Gaussian function applied to each of the pluralityof digital signals, wherein the second optical processing circuitryfurther comprises a second spatial light modulator for imprinting aninverse phase front of each received mode component on each the receivedmode components within the received single carrier signal to generatethe carrier signal having a planar phase front for each of the modecomponents within the carrier signal; and an elliptical core fiber fortransmitting the single carrier signal from the optical transmitter tothe optical receiver, the elliptical core fiber including an ellipticalcore have a major axis and a minor axis, wherein the elliptical corefiber further comprises a first stress rod located above the major axisalong the minor axis and a second stress rod located below the majoraxis along the minor axis.
 25. The system of claim 24 further includinga spatial filter for filtering the carrier signal from the secondspatial light modulator, wherein the spatial filter passes plane wavecomponents of the of the carrier signal and blocks other output modecomponents of the carrier signal.
 26. The system of claim 25, whereinthe second optical processing circuitry further measures a power outputof each mode component from the spatial filter.
 27. The system of claim26, wherein the measured power output of each mode component is used topopulate a mode crosstalk matrix.
 28. The system of claim 27, whereinthe optical transmitter launches the single carrier into the ellipticalcore fiber.
 29. The system of claim 24, wherein the at least one of theHermite-Gaussian function, the Laguerre-Gaussian function or theInce-Gaussian function comprises the Ince-Gaussian function.
 30. Thesystem of claim 24, wherein the first optical processing circuitryfurther combines each of the at least one of Hermite-Gaussian,Laguerre-Gaussian or Ince-Gaussian function applied plurality of digitalsignals into a single carrier signal using at least one of wavelengthdivision multiplexing or dense wavelength division multiplexing.
 31. Thesystem of claim 24, wherein the first optical processing circuitryfurther combines each of the at least one of Hermite-Gaussian,Laguerre-Gaussian or Ince-Gaussian function applied plurality of digitalsignals into a single carrier signal using a combination of polarizationmultiplexing, wavelength division multiplexing and mode divisionmultiplexing.